arXiv:1210.0809v1 [cond-mat.str-el] 2 Oct 2012May 25, 2018 22:30 Advances in Physics Hallreview7x Advances in Physics Vol. 61, No. 00, October 2012, 2{83 RESEARCH ARTICLE Hall e ect

May 25, 2018 22:30 Advances in Physics Hallreview7x

Advances in PhysicsVol. 61, No. 00, October 2012, 1–83

This is an Author’s Accepted Manuscript of an article to be published in Ad-vances in Physics [copyright Taylor & Francis], available after publication onlineat: http://www.tandfonline.com/

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Advances in PhysicsVol. 61, No. 00, October 2012, 2–83

RESEARCH ARTICLE

Hall effect in heavy-fermion metals

Sunil Naira†, S. Wirtha∗ , S. Friedemanna,b, F. Steglicha, Q. Sic and A. J. Schofieldd

aMax Planck Institute for Chemical Physics of Solids, Nothnitzer Str. 40, 01187 Dresden,

GermanybCavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, United KingdomcDepartment of Physics and Astronomy, Rice University, Houston, Texas 77005, USAdSchool of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT,

United Kingdom(Received 00 Month 200x; final version received 00 Month 200x)

The heavy fermion systems present a unique platform in which strong electronic correlationsgive rise to a host of novel, and often competing, electronic and magnetic ground states.Amongst a number of potential experimental tools at our disposal, measurements of the Halleffect have emerged as a particularly important one in discerning the nature and evolutionof the Fermi surfaces of these enigmatic metals. In this article, we present a comprehensivereview of Hall effect measurements in the heavy-fermion materials, and examine the success ithas had in contributing to our current understanding of strongly correlated matter. Particularemphasis is placed on its utility in the investigation of quantum critical phenomena which arethought to drive many of the exotic electronic ground states in these systems. This is achievedby the description of measurements of the Hall effect across the putative zero-temperatureinstability in the archetypal heavy-fermion metal YbRh2Si2. Using the CeMIn5 (with M =Co, Ir) family of systems as a paradigm, the influence of (antiferro-)magnetic fluctuations onthe Hall effect is also illustrated. This is compared to prior Hall effect measurements in thecuprates and other strongly correlated systems to emphasize on the generality of the unusualmagnetotransport in materials with non-Fermi liquid behavior.

Keywords: heavy fermion metals; Hall effect; quantum criticality; magnetotransport

Contents Page

1. Introduction 41.1. Significance of Hall effect to heavy-fermion systems 41.2. Contemporary issues in heavy-fermion systems 51.3. Classification of quantum critical points via Hall effect 61.4. Outline and scope 8

2. Basics of Hall effect and heavy-fermion systems 92.1. History of the Hall effect 92.2. The influence of magnetism: Anomalous Hall effect 102.3. Mechanisms contributing to the anomalous Hall effect 11

2.3.1. Skew scattering 112.3.2. Side-jump mechanism 122.3.3. Berry phase contributions 12

†Present address: Indian Institute of Science Education and Research (IISER), Dr Homi Bhabha Road,Pune 411008, India∗Corresponding author. Email: [email protected]


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Page

2.4. Hall effect and Fermi surface 132.5. Basic remarks on heavy-fermion systems 142.6. Anomalous Hall effect in heavy-fermion systems 16

3. Theoretical work on the Hall effect 19

3.1. Theoretical Overview 193.2. Key results from Boltzmann theory 203.3. Hall effect within Fermi liquid theory 223.4. Quantum critical heavy fermion metals 23

3.4.1. “Conventional” Hertz-Moriya-Millis quantumcriticality 23

3.4.2. Spin-fluctuation theory 243.5. Hall effect across Kondo breakdown quantum critical point 26

3.5.1. Quantum criticality in heavy fermion metals andjump of Hall coefficient 26

3.5.2. Crossover and scaling of the Hall coefficient at T 6= 0 28

4. Experimental aspects of Hall effect measurements in metals 294.1. Measurement techniques 294.2. Advanced aspects of Hall effect measurements 32

4.2.1. Single-field Hall experiments 324.2.2. Crossed-field Hall experiments 344.2.3. Realization of crossed-field Hall experiments 364.2.4. Comparing single and crossed-field Hall experiments 39

5. Hall effect and Kondo-breakdown quantum criticality 405.1. Hall effect evolution at the QCP in YbRh2Si2 405.2. Comparison to other candidates for Kondo breakdown 485.3. Hall effect and scaling behavior 51

6. Hall effect in systems with 115 type of structure 526.1. Influence of magnetic fluctuations on superconductivity 526.2. Interplay magnetism and superconductivity in CeM In5 546.3. Hall effect measurements on CeM In5 systems 56

6.3.1. Scaling relations in Hall effect 566.3.2. CeCoIn5 576.3.3. CeIrIn5 606.3.4. Comparative remarks 60

7. Comparison to Hall effect of other correlated materials 627.1. Copper oxide superconductors and related systems 62

7.1.1. Cuprates 627.1.2. Comparison of cuprates and heavy-fermion systems 677.1.3. Hall effect in oxy-pnictides and related systems 69

7.2. Other systems of related interest 707.3. Colossal magnetoresistive manganites 70

8. Summary 73

9. Appendix 74

Acknowledgement 75References 76


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1. Introduction

1.1. Significance of Hall effect in heavy-fermion systems

The heavy fermion metals represent an enigmatic class of strongly correlated elec-tron systems, in which the strong hybridization between the conduction electronsand localized f moments results in a Landau Fermi liquid (LFL) at low tempera-tures with heavily renormalized quasiparticle properties [1]. A dramatic manifesta-tion of the strong many body effects in these systems was uncovered in 1979 withthe discovery of superconductivity in CeCu2Si2 [2]. Predating the flurry of activityin the high temperature superconducting cuprates, this represented the first obser-vation of superconductivity in an inherently magnetic environment. It was followedby the observation of superconductivity in many heavy fermion metals [3]. It hasalso motivated the search for electronic (more specifically, magnetic) mechanismsfor superconducting pairing. The resurgence of research into the heavy fermion met-als is primarily due to current interest in continuous quantum phase transitions—azero-temperature instability which can be tuned by the use of a non-thermal controlparameter [4–6]. The presence of such a zero-temperature instability is often man-ifested in a large region of the experimentally accessible phase space, as has beenclearly demonstrated in many systems. The celebrated Landau-Fermi-liquid the-ory of conventional metals breaks down in the vicinity of such instabilities [5], andanomalous experimental behavior like a linear temperature dependence of the elec-trical resistivity [7], a non-saturating specific heat coefficient [8] and an apparentviolation of the Wiedemann-Franz law [9, 10] have been observed. This instabilityalso appears to be linked to the emergence of novel states of matter—for exam-ple superconductivity (as mentioned above) is often observed in the vicinity of aquantum phase transition [11]. A related development has been the growing real-ization that the physics of the heavy-fermion systems and the high-temperaturesuperconducting cuprates have much in common. What makes this so fascinatingis that unlike the heavy fermions metals, the parent high-transition-temperature(high-Tc) superconductors are Mott insulators. In this context it is surprising thatthe superconducting and normal state properties of some heavy fermion systemsare so similar to the high-Tc cuprates [12].

A microscopic explanation of the physics of the heavy-fermion systems calls foran understanding of how the Fermi surfaces of these complex systems evolve asa function of various experimental control parameters. Measurements of the deHaas–van Alphen (dHvA) effect and of angle resolved photo electron spectroscopy(ARPES) are amongst the most powerful tools for this purpose. In particular, dHvAeffect measurements demonstrated [13] the applicability of a description by stronglyrenormalized quasiparticles in UPt3. In general, however, extreme sensitivity todisorder along with prerequisites of very low temperatures and high fields havelimited the application of these techniques to the heavy-fermion systems. It is in thiscontext that measurements of the Hall effect [14]—a relatively simpler experimentaltechnique—has proven illuminating. This is in spite of the fact that the Hall effectitself is a complex quantity, especially in systems like the heavy-fermion systemswhich usually have more than one band crossing the Fermi level. Moreover, thecontribution from the anomalous Hall effect which results from skew scattering[15], is an important factor in materials comprised of an array of localized magneticmoments. However, it has been shown that at low temperatures the contributionfrom skew scattering is negligible in the heavy-fermion systems [16], and thus theexperimentally measured Hall voltage predominantly arises from the normal partof the Hall effect. Consequently, at these temperatures the Hall effect can providea crucial—albeit indirect—measure of the Fermi surface volume.


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Measurements of the Hall effect have also been extremely useful in revealing thenormal-state properties of heavy-fermion superconductors. For instance, the pos-sible presence of a pseudogap-like precursor state to superconductivity in CeM In5

system (M = Co, Ir) has been proposed. Comparison with prior results on the Halleffect in the superconducting cuprates reveals many striking similarities, indicat-ing that the non-Fermi-liquid physics probably affect the Fermi surfaces of thesediverse materials in a rather similar fashion. An anisotropic reconstruction of theFermi surface by the formation of “hot” and “cold” regions with different scatteringrates has been suggested, which then manifests itself in the form of two distinctscattering times that influence the resistivity and Hall effect in these materials indisparate manners [17, 18]. Recent measurements of the Hall effect in cuprates havereinforced its utility in uncovering the relation between quantum magnetism andsuperconductivity in these complex materials [19–21].

The theory of the Hall effect in strongly correlated systems remains an exceed-ingly difficult problem, in part due to the acute sensitivity of the Hall response tothe constitution and topology of the Fermi surface. Earlier calculations on met-als relied on approximating the Fermi surface to simple geometric constructionswithout taking into account the specific band structure of the materials under con-sideration [22, 23]. Unusual Hall behavior in the cuprates [24] triggered intenseactivity in this area, and besides a more generic geometric treatment [25], modelslike the nearly antiferromagnetic Fermi liquid [26] and the spin-charge separation[27] scenario, and others were employed to account for the experimental observa-tions. For the heavy-fermion metals, the Hall constant was predicted to allow for aqualitative distinction between different scenarios for quantum criticality [28]. TheHall effect and the resistivity have also been calculated using current vertex cor-rections (CVC) on a Fermi liquid in the presence of antiferromagnetic fluctuations,in an attempt to unify observations in the cuprates and the heavy-fermion metals[29].

1.2. Contemporary issues in heavy-fermion systems

The recent interest in heavy-fermion systems stems from their model characterin the investigation of quantum critical points. Although occurring at zero tem-perature, a quantum critical point (QCP) may lead to unusual properties up tosurprisingly high temperatures. We shall see that the Hall effect is a sensitive toolto explore the nature of a QCP.

Two main approaches are available to describe QCPs in heavy fermion metals(cf. Fig. 1). Conventionally, the order parameter notation is generalized by incorpo-rating quantum corrections. For the heavy-fermion systems magnetism is treateditinerantly giving rise to a spin-density wave (SDW) [4, 30, 31]. This approach re-lies on the fact that the composite quasiparticles formed by the Kondo effect stayintact at the quantum critical point, Fig. 1(a).

By contrast, more recent studies additionally incorporate quantum modes tobecome critical [32, 33]. In the case of the heavy-fermion systems the critical quan-tum modes are associated with the Kondo effect. When the Kondo effect becomescritical, the quasiparticles disintegrate at the QCP, Fig. 1(b). As a consequence,the Fermi surface is expected to jump at such an unconventional QCP. This is tobe contrasted with the conventional QCP for which a continuous evolution of theFermi surface is expected. Hence, the evolution of the Fermi surface is a fundamen-tal feature distinguishing the Kondo breakdown from the SDW scenario. In fact,Hall effect measurements were suggested to characterize QCPs [28]. The secondfundamental difference between the two scenarios is their scaling behavior. For the


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a) b)

TN

SDW

non-FL

LFL

δ

T

c δ

T0

TT

N

antiferro-magnetic order

*

LFLT

non-FL

LFL

δ

T

c δ

T0

Figure 1. Comparison of quantum criticality in a) the spin-density wave (SDW) and b) the locally criticalscenario. δ refers to an external control parameter for tuning the material to its QCP at δc. The hatchedareas indicate phase space regions within which the heavy quasi-particles are formed, whereas the arrowsillustrate the existence of local magnetic moments. In the local scenario (b) the quasi-particles themselvesbreak apart at δc. At finite temperatures the quasiparticles disintegrate within a crossover range indicatedby the T ∗ line. TLFL marks the temperature below which LFL behavior is found, whereas T0 denotes theonset temperature for initial Kondo screening.

3D SDW QCP an E/T 3/2 scaling is predicted for the spin dynamics, whereas anE/T scaling is expected at a Kondo breakdown QCP. The first example for an un-conventional QCP was identified in CeCu6−xAux by observation of an E/T scalingin inelastic neutron scattering measurements [34]. This Kondo breakdown scenariowill be further discussed in section 3.5.2. In the following we shall see how Halleffect measurements may shed light on the dynamics and on the scaling behavior.

1.3. Classification of quantum critical points via Hall effect

Hall effect measurements currently provide the best probe to study the Fermisurface evolution at QCPs. This is mainly due to the experimental difficultiesassociated with other probes: Photo electron spectroscopy on the one hand is—inthe context discussed here—limited to relatively high temperatures and moderateenergy resolution (∆E in the order of meV). Quantum oscillation measurements,on the other hand, can only be conducted in high magnetic fields which are oftenbeyond interesting energy scales, and require clean samples which often excludesthe investigation of composition-driven QCPs.

In order to distinguish whether the Fermi surface evolves continuously or dis-continuously one needs to have sufficient resolution of the control parameter usedto access the QCP. Widely employed tuning parameters in heavy fermion systemsare pressure, composition, and magnetic field. So far, however, only magnetic field-tuned QCPs appear to allow the high resolution required.

The QCP in CeCu6−xAux is tuned either by variation of gold content or bythe application of pressure. Probing a pressure- or composition-driven QCP withsufficient resolution was not yet able to show an abrupt Fermi surface change in thissystem by means of Hall effect measurements. The critical gold concentration ofxc = 0.1 implies that a considerable amount of scatterers are present which changethe Hall effect dramatically [35, 36]. The latter appear to lead to a dominance of theanomalous contributions to the Hall resistivity for all non-stoichiometric samples.Consequently, the normal contributions cannot be determined reliably and hence,information on the Fermi surface evolution is difficult to extract. This is also seenfrom the fact that no signatures are seen in the Hall resistivity when probing thefield-induced QCP in samples on the magnetically ordered side of the QCP [37].

Pressure reverses the effect of Au substitution in CeCu6−xAux. Consequently,the pressure-driven QCP is only accessible for samples with finite Au content.A Hall effect study on such a sample would presumably suffer from the stronganomalous contributions [36, 38]. Moreover, Hall effect studies under pressure are


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highly demanding. The fine tuning of pressure required to resolve a rapid change inthe Fermi surface is rarely achieved. This is seen for instance in the heavy fermionsystem CeRh2Si2 where the Hall coefficient was extracted as a function of pressure[39]. The few data points obtained in the pressure range up to 1.3 GPa provideonly limited insight. Here, the Hall coefficient seems to be sensitive to a changein the magnetic structure at 0.6 GPa. At the low temperature magnetic transition,however, the data appear to evolve smoothly despite the fact that de Haas–vanAlphen measurements indicate a strong Fermi surface reconstruction [40]. It wouldbe necessary to collect further data to draw a firm conclusion about the evolutionof the Fermi surface. We note that the sign change of RH observed in proximityto the critical pressure cannot directly be related to a severe Fermi surface changesince CeRh2Si2 features multiple bands [40], cf. discussion in section 2.4.

For the material CeRhIn5 a drastic change of the Fermi surface at a criticalpressure of about 2.3 GPa was observed by de Haas–van Alphen measurements[41] which has been ascribed to a breakdown of the Kondo effect at an underlyingQCP [42]. It is to be expected that the Hall coefficient will correspondingly showa drastic change across the critical pressure. Existing Hall measurements withinthe zero-field limit and at relatively high temperatures (above 2 K) have alreadyrevealed a rapid change of the Hall coefficient across the critical pressure; this rapidchange is likely associated with both the Kondo physics and the QCP, because it isabsent in both LaRhIn5 and CeCoIn5 [12]. In order to determine the precise natureof the evolution of the Hall effect across the critical pressure, it is important tocarry out measurements at low temperatures (0.4 K or below, where the electricalresistivity shows a T 2 dependence [43]) and at magnetic fields where superconduc-tivity is suppressed. It should also be stressed that the precise evolution of the Hallcoefficient across the QCP needs to be studied with care, given the discrete stepsin pressure used as the control parameter.

A canonical SDW QCP is realized in pure and V-doped Cr [44]. Hall effect mea-surements under pressure conducted on these materials represent the rare exampleswhere the Hall coefficient was measured for a large series of pressure points [45, 46],see Fig. 2. These measurements were conducted at 5 and 0.5 K, respectively, i.e.at temperatures which are small compared to the natural temperature scales ofthese systems. The measurements reveal a smooth crossover of the Fermi surfacewhen the magnetic order is suppressed, a behavior which is expected theoretically[28, 47, 48]. The Fermi surface of a SDW state is reconstructed from that of theparamagnetic state through a band folding, which is more pronounced for systemslike Cr whose Fermi surfaces are nested. However, if the SDW order parameter isadiabatically switched off the folded Fermi surface is smoothly connected to theparamagnetic one. As a result, the Hall coefficient does not show a jump as longas the nesting is not perfect [48]. So far, no information was extracted regardinghow this crossover evolves with temperature; it would be interesting to see if thefinite crossover width persists in the extrapolation to zero temperature as expectedfor a SDW scenario. However, we wish to point out that these pressure measure-ments were very extensive and it would be an even larger effort to implement thetechniques used to the low temperatures needed to study heavy fermion systems.

The remaining common tuning parameter, magnetic field, appears to be theobvious choice to achieve high resolution. In case of a magnetic-field driven QCP,however, the non-zero critical field can give rise to a small discontinuity in themagnetotransport coefficients even in the case of an SDW QCP [50]. Approachingthe QCP, the energy scale associated with the vanishing order parameter becomessmaller than the scale associated with the Lorentz force, leading to a non-linearityin the system’s response to the Lorentz force. This is reflected in a breakdown of the


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Figure 2. Inverse Hall coefficient of pure Cr (lower panel) and Cr0.968V0.032 (upper panel) as a functionof pressure. In the upper panel, ∆p denotes the external pressure for Cr0.968V0.032, or an effective pressurefor samples in which the V-concentration is different from 0.032 [46]. In the lower panel, the result is shownonly in the vicinity of the critical pressure [45] so that the two panels cover the same pressure extent. Thedashed lines label the critical pressure. Reproduced with permission from S. Friedemann et al., Journal ofPhysics: Condensed Matter 23 (2011) 094216 [49]. Copyright (2011) IOP Publishing Ltd.

weak-field magnetotransport. The linear field dependence of the magnetoresistancein Ca3Ru2O7 was taken as indication thereof [51]. Moreover, the breakdown ofweak-field magnetotransport may lead to a jump in the Hall coefficient which,however, is likely to be very small. In fact, for the case of the cuprates mean fieldtheory predicts that the anomaly in the Hall response at the critical doping levelis negligibly small even though the SDW gap is large [52]. Disorder may smear thejump of the Hall coefficient into a smooth crossover.

For the above-mentioned pressure-driven SDW QCP of pure and V-doped Cr[45, 46], the Hall coefficient (and the resistivity) exhibits a smooth crossover nearthe QCP the width of which does not track the strength of disorder (Fig. 2). Thisbehaviour is in contrast to the scenario of a breakdown of the weak-field limit. Forthe field-driven QCP of YbRh2Si2, the effect of non-linear response to the Lorentzforce is expected to be negligible, given that at the critical field, ωcτ as defined ineq. (3) is very small (on the order of 0.01 and 0.002 for the samples described inRef. [53]). Furthermore, it is completely avoided by the crossed-field Hall setup,see section 4.2.2.

1.4. Outline and Scope

In this article, we give an overview of experimental and theoretical work on the Halleffect in the heavy fermion metals. We start with a brief description of the basics ofthe Hall effect, and an introduction to the anomalous Hall effect in section 2. Subse-


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quently, we introduce the heavy-fermion metals and describe the ingredients whichgo into the formation of the heavy-fermion fluid. A description of the anomalousHall effect is carried over from the earlier section, and a more microscopic descrip-tion of this effect is provided. The relevance of Hall effect measurements in theinvestigation of these systems is stated, and the use of the Hall effect in identifyingsome transitions and crossovers observed in the low-temperature phase diagrams ofthe heavy-fermion systems is described. Section 3 provides an account of theoreticalwork on the Hall effect in strongly correlated electron systems. This incorporatesmodels ranging from the earlier geometric methods to contemporary ones wherethe details of band structure are explicitly considered. A brief description of theexperimental aspects of Hall effect measurements are provided in section 4. Spe-cial mention is made of the crossed-field Hall experiments, where a tuning field isintroduced in addition to the magnetic field used for generating the Hall response.The influence of strong magnetic fluctuations on the physical properties of theheavy-fermion systems is well documented. Section 6 describes the signatures ofthese magnetic fluctuations as seen in Hall effect measurements. Special mentionwill be made of the CeM In5 (M = Co, Ir or Rh) family of compounds where acomplex interplay between superconductivity and magnetism has been observed.Section 5 deals with the investigation of quantum critical phenomena – a topic ofcontemporary interest. The utility of Hall effect measurements in the investigationof quantum criticality will be outlined, and experimental work in this area is re-viewed. In section 7, we compare the Hall effect in the heavy-fermion systems withobservations in other strongly correlated electron systems, with emphasis on themanganites, doped Mott insulators and the copper oxide superconductors. Section8 sets out some open questions.

One cannot hope to be comprehensive in this vast subject, so we have omittedseveral issues entirely. These include the heavy-fermion transition metal oxides aswell as the mixed state of the heavy-fermion superconductors. Since only a verybrief description on early work in metals is included in section 2, we refer the readerto earlier reviews [14, 54, 55] which deal exclusively with the Hall effect in metalsand alloys.

2. Basics of Hall effect and heavy-fermion systems

2.1. History of the Hall effect

Unconvinced by a passage from Maxwell’s treatise on Electricity and Magnetismwhich stated that “the path of a current through a conductor is not permanentlyaltered by a magnetic field”, Edwin H. Hall, in 1879, set about investigating theaction which a magnetic field would have on such a current. Using experimentson a gold leaf held between the poles of an electromagnet, he observed that whena magnetic field is applied perpendicular to the direction of a current flow, anelectric field is generated perpendicular to both, the direction of the current, andthe direction of the magnetic field [56]. This resultant electric field could thenbe detected using a sensitive galvanometer. Hall also realized that the product ofthe current through the specimen and the strength of the magnetic field, whendivided by the current through the galvanometer, was reasonably constant. Incurrent notation, this implies that VH ∝ BI, where B and I refer to the magneticflux density and the current flowing through the specimen, respectively, and VH

denotes the transverse (Hall) voltage.Intuitively, this can be seen as a direct consequence of the Lorentz force acting on

the electron current flowing through the solid. Consider a rectangular specimen of


10

Figure 3. Schematic setup for the measurement of the Hall effect.

width b and height d as shown in Fig. 3 and assume a current Ix of charge carriersq with density n flowing uniformly along its length with velocity vx. When Bz isapplied perpendicular to the direction of the current, the electrons are deflected bythe Lorentz force towards the edge of the specimen. This transient motion buildsup a charge at (and thus an electric field Ey ≡ EH between) the edges of thesample, which in turn impedes further electrons from accumulating at the edges.A stationary state is achieved when the deflection of charge carriers due to theLorentz force is balanced by the force resulting from the transverse electric field:EH − (vxBz) = 0. With Ix/bd = nqvx one finds for the Hall voltage Vy = EH/bwithin the simple free electron model:

Vy =1

nq

IxdBz = RH

IxdBz (1)

The constant of proportionality RH = 1/nq is the so-called Hall coefficient. Ob-viously, Hall measurements can reveal information on the type of charge carriers.This was soon realized with the observation of a positive RH in some metals. Eq.(1), however, is strictly valid only if the material can be described by a simpleone-band picture. In more complex materials, more then one band may exist atthe Fermi energy EF and hence, contributes to the conductivity. In these cases,the charge carrier concentration as determined from Hall measurements has tobe considered an effective one, neff , and the interpretation of the results can beconsiderably complicated, see e.g. [14] and section 2.4.

2.2. The influence of magnetism: Anomalous Hall effect

The Hall effect in metals is a direct consequence of the breaking of time reversalsymmetry, in this case by the application of an external magnetic field H. It wasrealized very early on that the Hall effect could possibly arise even in the absenceof an external field, as long as time reversal symmetry is broken. Early experimen-tal insight was provided by measurements on ferromagnetic elements and alloys,where unlike that observed in simple diamagnetic metals, a pronounced nonlinear-ity in the field dependence of the Hall signal was observed [57–59] (paramagneticmaterials may exhibit small nonlinearities). The Hall voltage was not only seento be proportional to the magnetization M , but also appeared to reproduce theirreversibilities observed in the magnetization curves [60–62]. This additional con-tribution to the ordinary, or normal, Hall effect was termed the extraordinary, oranomalous, Hall effect (AHE). At fields larger than that required to drive the mag-netization of the material into saturation, the Hall effect continued to increase,albeit at a much slower rate.

Empirically, the measured Hall resistivity in ferromagnets could thus be written


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µ0M

S

RSµ

0M

S

B

ρρρρxy

µ0M

S(R

0+ R

S)

Figure 4. Schematic of the typical behavior of the Hall resistivity %xy in a magnetic field.

as

%xy = R0B +RSµ0M (2)

where R0 and RS refer to the normal and the anomalous Hall effect contributions,respectively. Fig. 4 shows a schematic of the typical behavior of the Hall resistivityin a metal with appreciable magnetization as a function of applied field. Two dis-tinct, linear regimes with different slopes are clearly discernible. In the first one, theslope equals R0 +RS. The second regime lies above the technical saturation field,and here the smaller slope corresponds to R0 only. The straight extrapolations ofthese two regimes intersect at B = µ0MS where MS is the spontaneous magneti-zation. Though the empirical formula (2) assumes that the anomalous Hall effectis directly proportional to the magnetization, the observed phenomenon could notbe simply explained in terms of the internal field of the ferromagnetic system. Anon-trivial temperature dependence and even a change of sign of the Hall signal atlow temperature in some systems, warranted the use of more sophisticated modelsto explain the experimental data (see, e.g., Ref. [63]). It is now understood thatthe extent and dependencies of the AHE arise as a consequence of the spin-orbitcoupling, although some of the microscopic details continue to remain investigated.In what follows three basic mechanisms which have been used to account for theanomalous Hall effect, namely “skew scattering”, the “side-jump” mechanism andthe “Berry phase” induced Hall signal, are described in more detail. For early re-views of the anomalous Hall effect, the reader is referred to Refs. [14, 64] whereasmore contemporary reviews on the anomalous Hall effect are provided in Refs. [65–67].

2.3. Mechanisms contributing to the anomalous Hall effect

2.3.1. Skew scattering

First postulated by Smit, skew scattering refers to the phenomenon of electronstraversing in a plane perpendicular to the applied magnetic field being scatteredasymmetrically from a magnetic impurity potential [15]. Considering a stream ofelectrons traversing along a plane, Smit argued that in a perfectly periodic lat-tice, the transverse polarization resulting from the spin-orbit interaction is fullycompensated by the periodic electrostatic forces. However, in a real crystal whereperiodicity is imperfect, the compensation near this impurity potential is incom-plete. This can result in a finite force which accelerates electrons perpendicular tothe direction of electron flow. It was also suggested that for the skew scatteringmechanism the Hall resistivity should vary linearly with the conventional electri-cal resistivity (%xy ∝ %xx). Since the early days of investigations on Kondo lattice


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systems, this mechanism has been widely cited to be primarily responsible for theanomalous Hall contribution observed in these systems. Here, the magnetic im-purities which are immersed in the sea of conduction electrons act as impuritypotentials for electron scattering. These magnetic impurities can be polarized byapplying an external magnetic field, thus deflecting the charge carriers preferen-tially in one direction giving rise to a large anomalous contribution to the Halleffect.

2.3.2. Side-jump mechanism

The observation of the Hall resistivity varying quadratically as a function of theresistivity (%xy ∝ %2

xx) in iron [68] and some alloys [69] is inconsistent with thepredictions of the simple skew scattering mechanism, where a linear relation isexpected. This led Berger to propose an alternative explanation for the anomalousHall effect which was termed the side-jump mechanism [70]. Berger argued thatwhen an electron is scattered by an impurity potential (or a phonon), the trajectoryof the electron is shifted sideways by the action of the spin-orbit coupling. Thisabrupt jump takes place in a direction perpendicular to both the initial electrontrajectory as well as the magnetization. Thus, unlike the skew scattering mechanismin which the electron can propagate in a direction perpendicular to the originaldirection after scattering, here the electron is simply displaced sideways by a smalldistance. This mechanism which was suggested to be independent of the density ofimpurities could be used to account for the observed %xy ∝ %2

xx behavior. Since thisdisplacement is characteristically smaller than the electronic mean free path, thecontribution arising due to the side-jump mechanism is also typically much smallerthan that stemming from skew scattering. However, if the mean free path becomesof the order of this displacement, then this mechanism could dominate. Therefore,consequences of the side-jump mechanism are more probable to be observed atrelatively higher temperatures or in samples with appreciable degree of disorder.

Recent calculations using density functional theory (DFT) showed that the side-jump contribution to the anomalous Hall effect can directly be computed from theelectronic structure of a pristine crystal, and good agreement with experimentaldata was observed in the cases of some ferromagnetic elements and alloys [71].

2.3.3. Berry phase contributions

Unlike the skew scattering and side-jump mechanism which deal with scatteringfrom impurity potentials and thus, are clearly extrinsic in nature, the Berry phasecontribution to the AHE is an intrinsic geometric contribution. The nomenclature isdue to M. Berry, who first pointed out that a quantum system which is adiabaticallytransported along a closed loop acquires a phase which depends purely on thegeometry of the loop [72]. Interestingly, though the realization of the existence ofthis geometric phase and its importance is relatively recent, Karplus and Luttingerhad already in 1954 discovered the probably earliest instance of a Berry phase ina solid in an attempt to explain the origin of the anomalous Hall effect [73, 74]. Incalculating the electron transport in a system with broken time reversal symmetry,they uncovered a term which mimicked the influence of a magnetic field, thusgiving rise to a dissipationless transverse current. The Berry phase contributionhas now been observed in some systems such as spinels [75] and pyrochlores [76]and continues to be a field of intense activity especially in the field of spintronics[77]. However, its relevance—if any—to the heavy-fermion systems remains to beevaluated.

We note that the side-jump mechanism could also be considered as a consequenceof the Berry phase contribution in samples with moderate impurity concentration.This leads to the same dependencies on resistivity, %xy ∝ %2

xx, in case of both


13

mechanisms [67].

2.4. Hall effect and Fermi surface

Eq. (1) and all considerations based on it (including RH = 1/nq) rely on Drude’smodel of a free electron gas. However, already in simple metals like Al its as-sumptions no longer hold and, e.g., RH depends strongly on magnetic field [78].In general, RH in metals strongly depends on the actual band structure and theparticular shape of the Fermi surface onto which the electrons traverse.

Before continuing we shall introduce a useful quantity, the so-called Hall angleθH. It basically describes the angle between the total electric field and the current

tan θH =%xy%xx

∝ ωcτ = µB , (3)

where ωc is the cyclotron frequency, τ is the relaxation time and µ the mobility. Forsmall magnetic fields, the Hall angle can be considered as the angular deviationof the electron’s motion within τ . In high magnetic field and within the simpleDrude picture, θH approaches π/2. According to eq. (3) the Hall angle providesa direct measure of ωcτ and hence, the mobility µ. Note that for large values ofωcτ = eBτ/meff not only high magnetic fields but also low temperatures and singlecrystals of sufficiently good quality are required.

If all occupied electronic levels fall on closed orbits the high field limit of the Hallcoefficient again simplifies to RH = 1/nq (a similar consideration holds for holes).This is no longer valid if one (or more) orbit is open in kkk-space. Depending ondirection of this orbit in real space the high field limit of θH can drastically deviatefrom π/2 and, more generally spoken, the Hall effect can become anisotropic withrespect to the sample’s crystallographic orientation. Of course, in the small-fieldlimit (B → 0) the precise nature of the orbits is insignificant.

In the majority of metals there is more than one band crossing the Fermi energyEF. In consequence, all these bands, electron- or hole-like, contribute to the re-sulting Hall effect. Although in principal the voltages generated by the individualbands simply sum up, the resulting expressions quickly become cumbersome (ingeneral, the Hall resistivity is a matrix element obtained by inverting the conduc-tivity tensor, see below). For two bands (the expression can easily be generalizedto more than two bands) one finds in the low-field limit [14]

RH =1

e σ2t

[σ2

1

n1+σ2

2

n2

]where σt = σ1 + σ2 (4)

and the indices refer to the individual bands. If an effective carrier concentrationis introduced as

1

neff=

1

σ2t

[σ2

1

n1+σ2

2

n2

](5)

than the expressions for the single-band case are recovered. Each band is char-acterized by two parameters: its carrier concentration ni and its mobility µi (orrelated quantities). Therefore, an analysis of measured Hall effect data even fortwo-band conductors is complicated as has clearly been demonstrated for CrO2

[79]. For more than two bands any quantitative analysis is severely hampered bythe numerous free parameters. Only in the high-field limit, i.e. for B → ∞, is a


14

simple relation within the two-band model recovered, RH = 1/e(n1 + n2). Experi-mentally, however, it is difficult to ascertain that this condition is met, and so forall bands involved.

A peculiar case arises for two-band conductors in which an electron and a holeband exhibit equal carrier concentrations, so-called compensated metals. In thiscase, there is a dominating contribution to %xy(B) which goes as B2 [80, 81].The latter has been demonstrated to hold by high-field experiments on the heavyfermion metal UPt3 [82]. This aspect will be important for the discussion of theHall effect on CeM In5 systems, sections 6.3.2 and 6.3.3.

2.5. Basic remarks on heavy-fermion systems

The assumption of non-correlated, i.e. essentially non-interacting, electrons is wellestablished throughout many areas of solid state physics. An excellent examplefor this is semiconductor physics which is typically understood in terms of non-interacting electrons. However, strong correlations between the electrons could offernew concepts and applications. Interest is fueled by the fact that the propertiesof the whole ensemble of interacting entities (in our case the electrons) may leadto new organizational principles or low-lying excitations neither related to norexpected from the properties of the individual constituents. The occurrence of suchnew principles are referred to as emergent behavior. Typical examples here rangefrom superconductivity, colossal magnetoresistance, to the fractional quantum Halleffect.

Originally formulated in an effort to explain the bulk properties of 3He, Lan-dau’s Fermi liquid theory [83] has found remarkable success in explaining the lowtemperature properties also of many materials that show strong electronic corre-lations. A key ingredient here is the notion of “quasiparticles”, which refer to low-lying excitations that have a one-to-one correspondence in terms of their quantumnumbers with the original particles, which in the case of metals are the conductionelectrons. Moreover, the Landau Fermi liquid theory could also predict the temper-ature dependencies of experimentally measurable quantities at low temperatures.For instance, the contribution of electron-electron scattering to the electrical resis-tivity % varies as T 2, the spin susceptibility χ is temperature independent, and thespecific heat divided by temperature, C/T , is a constant. Based on this concept,the occurrence of low temperature superconductivity in simple metals can well beaccounted for by the BCS theory [84], where phonons provide the glue that holdstogether electrons into Cooper-pairs.

The heavy fermion metals are a remarkable example of the robust nature ofthe Fermi liquid state. This is especially the case if these materials contain cer-tain elements with partially filled f electron shells, like Ce, Yb or U. At relativelyhigh temperatures, these incompletely filled f shells give rise to localized atomicmoments which interact only weakly with the sea of conduction electrons thatengulf them: the conduction electrons experience a weak energetic preference toalign their spins antiparallel to the total spin of the open f -shell. However, whenthe temperature is reduced, this antiferromagnetic interaction between the local-ized f -spins and the conduction electron spins becomes continuously stronger, andthis radically influences the low-energy ground states of the system. These strongcorrelations are manifested in the form of anomalous contributions to various ther-modynamic and transport properties. Moreover, the f electron moments in this lowtemperature state are seen to be appreciably smaller than their high temperaturevalues and, eventually, disappear in static properties for T TK, the so-calledKondo temperature—a consequence of the “Kondo effect”. The resistivity in the


15

T

JJc

TN

0

TK∝ 1g(EF)

exp(− 1J g(EF)

)

TRKKY∝ J2g(EF)

Figure 5. Phase diagram of heavy fermion metals as inspired by Doniach [89]. Here, J refers to the Kondoexchange coupling between the spins of the f and conduction electrons and g(EF) to the density of states(DOS) at EF.

high temperature limit is large as a consequence of spin flip scattering, and a dra-matic reduction is seen as one enters the low temperature coherent regime. If notinterrupted by the onset of magnetic order, the heavy fermion metal can then set-tle into a LFL regime with strongly renormalized quasiparticles having effectivemasses meff up to three orders of magnitude larger than the bare free electronmass.

The low temperature electronic ground states of these systems are thus dictatedby the intricate interplay between two competing phenomena: the Kondo effect[85], which screens the local moments and promotes the formation of a nonmag-netic ground state, and an indirect exchange interaction, the so-called Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [86–88], which couples the (partiallycompensated) f -spins via the conduction electrons. The Doniach phase diagram[89] provides a useful illustration of how the electronic ground states of these sys-tems evolve as a function of the coupling (or the hybridization) between the felectrons and the conducting electrons, as is schematically shown in Fig. 5. At lowantiferromagnetic Kondo coupling (J) values, the RKKY interaction (character-ized by the energy scale TRKKY ∝ J2) dominates the Kondo screening temperature(TK ∝ e−1/J) and the system orders magnetically. At high values of J , on the otherhand, the Kondo screening prevails and the system condenses into a sea of highlyinteracting quasiparticles with LFL characteristics. The resurgence in the field ofheavy fermions is predominantly due to current interest in quantum critical be-havior. This behavior is observed if the material undergoes a continuous phasetransition at absolute zero temperature; in the scenario just discussed such a tran-sition can take place from a magnetically ordered phase into a heavy LFL one,with the latter representing an ordered state in kkk-space. Within the framework ofDoniach’s description of the Kondo lattice, systems for which such phenomena arelikely to occur are those where the Kondo effect and the RKKY interaction havecomparable energies, i.e., for intermediate values of J . A quantum phase transitioncan then be brought about by a well-directed change of a non-thermal experimen-tal (“control”) parameter such as chemical doping, pressure or magnetic field [4].While the former two can directly influence the value of J (e.g. via changed latticeparameters) the latter may suppress antiferromagnetic order and, in particular fortransverse field tuning, even ferromagnetic order. Even though such a quantumphase transition at T = 0 is not directly accessible to experiment it affects thefinite temperature properties of the material if investigated sufficiently close to theQCP at which the continuous phase transition takes place in the temperature–


16

control parameter diagram [5]. The Landau Fermi liquid theory of conventionalmetals breaks down in the vicinity of such instabilities, and anomalous experimen-tal behavior like a power-law temperature dependence of the electrical resistivity(%(T )− %(T =0) ∝ T ε where 1 ≤ ε < 2) [7], a diverging specific heat coefficient [8]and even an apparent violation of the Wiedemann-Franz law [10] have now beenobserved in heavy fermion systems.

Experimentally, quantum critical behavior and unconventional superconductivityare often found in close vicinity in phase space in this class of materials. In addi-tion to the compound CeCu2Si2 [2], the compounds in which the phenomenon hasbeen observed include other Ce [90, 91], but also U [3, 92–95] and more recentlya Yb [96] based system. The observation of superconductivity in these systemswhich are comprised of a dense array of magnetic atoms is striking since the con-ventional BCS theory predicts that the presence of even very small amounts ofmagnetic impurities is highly detrimental to the formation of the superconductingcondensate. The fact that superconductivity can exist in an inherently magneticenvironment has forced researchers to think beyond the conventional scenario ofphonon-mediated superconductivity. The driving force for Cooper pair formationin these heavy fermion metals is believed to be electronic (or more specifically,magnetic) in origin [6, 97–100]. In case of the compound CeCu2Si2, an inelasticneutron scattering study of the magnetic excitation spectrum provided indicationsfor superconducting pairing resulting from antiferromagnetic excitations in thisprototypical heavy-fermion compound [101]. An earlier dramatic manifestation ofthis aspect is the superconductivity observed when a continuous magnetic phasetransition is suppressed to absolute zero temperatures, in other words in the vicin-ity of a QCP [11]. This has also reinforced the connection of heavy fermion systemswith the high transition temperature superconducting cuprates. For the latter, oneof the competing scenarios is that a QCP lies beneath the superconducting domeand is responsible for both superconductivity as well as the strange metallic (non-Fermi-liquid like) behavior observed in an appreciable region of the experimentallyaccessible phase space [102].

The formation of the heavy fermion state and superconductivity in these mate-rials has been a subject of extensive investigations in recent years, the details ofwhich are beyond the scope of this article. We refer the reader to some compre-hensive reviews devoted to both the experimental and theoretical aspects of thesesystems [1, 5, 103–105].

2.6. Anomalous Hall effect in heavy-fermion systems

Though the skew scattering mechanism has been used extensively in trying to ac-count for the observed Hall effect in heavy-fermion systems, a few caveats needto be borne in mind. Firstly, the skew scattering mechanism was proposed forsystems with ferromagnetic order, i.e., where the time reversal symmetry is irre-vocably broken. This criterion is not strictly met in the heavy-fermion systems,although a lack of time reversal symmetry can be induced by the application ofa magnetic field. Secondly, these scattering theories considered the simple case inwhich the same type of electrons are responsible for both magnetism and electri-cal conduction. In the heavy-fermion systems it is clear that the magnetism arisesfrom localized (partially screened) f electrons, and are thus of a different originfrom the conduction s, p or d electrons. A more relevant model here is that byKondo [106], who considered a situation in which d electrons were localized in asea of conduction (s shell) electrons. These localized spins can be disordered bythe influence of thermal fluctuations and, in turn, can then act as scattering po-


17

skew scatteringincoherent

+ skew scattering by residual defects)background (ordinary Hall effect

skew scattering

state

RH

T

skew scatteringby fluctuationsabout thecoherent

intrinsic

~ Tcoh

Figure 6. Modeling of the Hall effect in heavy-fermion compounds. Reprinted figure with permissionfrom A. Fert and P.M. Levy, Physical Review B 36, 1907 (1987) [16]. Copyright c© (1987) by the AmericanPhysical Society.

tentials. Thus, the non-periodicity originates in this case from thermal fluctuationsof the local spins, and not primarily from the introduction of impurities. Thoughthis s−d spin-spin interaction itself did not give rise to skew scattering, includingthe spin-orbit interaction of the d electrons within the magnetic ions did result inskew scattering, and thus an anomalous contribution to the Hall effect. A relatedtreatment of the Kondo model was presented by Maranzana [107] who considereda (d-spin–s-orbit) interaction which described the force acting on moving electronsas a consequence of the magnetic field produced by the magnetic d ions. This wasused to reproduce the temperature dependence of the anomalous Hall effect insome ferromagnets as well as the abrupt decrease in the Hall coefficient at theantiferromagnetic transition temperature (TN ), though the calculated values weresmaller than the experimentally measured ones. A similar model was also appliedby Giovannini to explain the experimentally observed anomalous Hall effect in somedilute magnetic alloys [108].

Considerable theoretical insight into the Hall effect in the heavy-fermion systemsoriginates from the early work by Fert [109] who considered the skew scattering inalloys containing Ce impurities. Using a Coqblin-Schrieffer interaction [110], whichconsiders a 4f1 configuration of Ce and its interaction with the conduction (s)electrons, a reasonable agreement with the low-field experimental data of someLa-Ce alloys was established. Subsequently, a model pertaining to a Kondo latticesystem was used [111, 112] in an effort to explain the early experimental signa-tures observed in the heavy-fermion metals. Here, the ground state was modeledby a degenerate effective f resonance level with an unquenched orbital moment.Application of a magnetic field lifts the degeneracy of the resonance level due toZeeman splitting, and gives rise to skew scattering of the band electrons. It is tobe noted that both these models were effectively meant to be applied to the in-coherent Kondo regime (T TK) where the Kondo ions scatter independent ofeach other; they were not strictly valid for the low temperature regime (T TK),where a coherent heavy-fermion band condenses out of the local moment landscape.Nevertheless, their results could—at least qualitatively—be extended down to thecoherent regime. Combining both these approaches, a generic interpretation of theHall effect in the heavy-fermion systems was put forward [16] which is schematicallyshown in Fig. 6. It was concluded that the skew scattering contribution increases asthe temperature is lowered and attains a maximum at the temperature Tcoh wherespatial coherence sets in. At even lower temperatures this contribution rapidly fallsdown to zero. In the high temperature regime, the Hall effect arising due to skewscattering is proportional to the product of the magnetic susceptibility (χ) and themagnetic contribution to the resistivity (%M ). Well below Tcoh, the Hall effect isprimarily composed of the ordinary Hall effect, in addition to a small contribution


18

from defects or impurities. This inference is striking, for it suggests that at verylow temperatures the measured Hall effect is an intrinsic quantity and thus can beeffectively used to monitor the evolution of the Fermi surface. This has now beenexploited in the investigation of quantum critical phenomena, an aspect which willbe dealt in detail in section 5.

Since the anomalous Hall effect exhibits the just described maximum at the onsetof the Kondo coherence, it is not surprising that a large positive maximum followedby a precipitous drop was observed in the measured Hall signal of a number of Ceand U based heavy-fermion systems [113–119]. In these reports, this drop in themeasured Hall response was usually ascribed to the formation of a coherent bandwhich drastically reduces electron scattering. In some systems, such as CePd3,CeBe13 and CeCu6, the drop in the Hall coefficient was accompanied by a changeof its sign. This change of sign of RH was accounted for by the models describedabove in spite of their limited validity in the low temperature (T TK) regime. Forinstance, Ramakrishnan and coworkers had predicted [112] that the Hall coefficientcan be written as

RH = R0 + gµB|α| % sin(φ+ δ2)/ sin δ2 . (6)

Here g is the gyromagnetic ratio of the f electrons, µB is the Bohr magneton andδ2 refers to the phase shift due to scattering in the l = 2 channel. In addition,α is proportional to χ1(1 − χ1T ) where χ1 refers to the reduced susceptibility.If the resonant scattering in the l = 3 channel at low temperatures gives riseto a phase shift δ3, then φ changes from −π to −2δ3 as one traverses from thehigh temperature (T TK) regime to the low temperature (T TK) one. Ac-cording to this model, in the high temperature regime, φ = −π, and thus theanomalous Hall contribution is always positive. In the low temperature regime onefinds φ = −2δ3 and—interestingly—here the Hall effect can have either sign. How-ever, based on measurements on the Ce1−xYxPd3 alloys, it was suggested that thischange of sign in RH is solely associated with the onset of coherence [120]. This wasdemonstrated by the fact that in compounds with finite Y substitution, in whichno low-temperature coherent state is attained, the Hall coefficient RH decreasedmonotonically as a function of decreasing temperature. Qualitatively similar be-havior was also observed in the system Ce(Pd1−xAgx)3 [113]. These results are incontrast to those observed for the undoped compound CePd3, where the formationof a low-temperature coherent state is accompanied by a sign change in RH. Theexperimental signature of the onset of coherence appeared to be much sharper inthe Hall effect if compared to that typically seen in resistivity measurements [121].This difference is exemplary demonstrated for the system CeCu6 in Fig. 7.

In many heavy-fermion systems, the onset of the low-temperature coherent stateis interrupted by the emergence of long-range (antiferro-)magnetic order. In thesystem U2Zn17, it was shown that RH increases abruptly at the onset of the anti-ferromagnetic order [122]. It was suggested that this change reflects the opening ofa gap at the Fermi surface due to magnetic ordering. In the system CePtSi, this in-crease was seen to be more dramatic, with the Hall constant increasing by a factorof 15 on entering the antiferromagnetically ordered regime [119]. Moreover, belowthe Neel transition temperature TN , the Hall resistivity was highly nonlinear, andalso exhibited signatures of a possible metamagnetic transition. However, the re-sistivity was almost constant around TN , thus prompting the authors to concludethat the observed features cannot be accounted for on the basis of a Fermi surfacemodification alone, which would have affected the resistivity and Hall measure-ments in similar fashions. Thus, irrespective of the interpretation of the observed


19

-2

-1

0

1

0.1 1 10 100

-100

-50

0

50

ρ (1

0-6 Ω

cm

)

T (K)

CeCu6

RH (

10-3 c

m3 /C

)

Figure 7. Comparison of the resistivity % ( and right scale) and the Hall coefficient RH ( and leftscale) at the onset of the coherent state in CeCu6. Reprinted figure with permission from T. Penney etal., Physical Review B 34, 5959(R) (1986) [121]. Copyright c© (1986) by the American Physical Society.

Hall coefficient across the antiferromagnetic transition, the sensitivity of this mea-surement in tracking such transitions is beyond doubt. Moreover, it appears to belarger than that commonly observed in resistivity measurements. This has beenreinforced by measurements on the heavy-fermion system URu2Si2 where a tran-sition at 17.5 K has been the focus of extensive investigations. The challenge hereis to reconcile the large entropy release and the accompanying gap in the mag-netic excitation spectrum with the anomalously small value (∼ 0.03 µB/U) of themagnetic moment, giving rise to suggestions that a “hidden order” coexists andprobably competes with the antiferromagnetic order in this system [123]. Earlyinvestigations in URu2Si2 revealed that the Hall coefficient RH exhibits a sharpanomaly at this transition [124]. An analysis based on the model by Fert and Levy[16] has been used to extract a carrier density of about 0.04 holes/U above thetransition, but is an order of magnitude smaller below the transition [125]. In com-bination with Nernst effect measurements, it was thus inferred that this transitionis accompanied by a reconstruction of the Fermi surface into small high-mobilitypockets, resulting in an abrupt increase in the entropy per itinerant charge carrierand a decrease of the scattering rate [126]. Recently, Hall effect measurements havebeen extended up to magnetic fields of the order of 45 T and revealed a cascadeof field induced transitions accompanied by changes in the Fermi surface topology[127, 128].

3. Theoretical Work on the Hall effect

3.1. Theoretical Overview

While the conceptual basis of the Hall effect is accessible enough for it to be cap-tured in a simple sentence—a manifestation of the perpendicular Lorentz forceon moving charges—the interpretation of Hall measurements can be considerablymore challenging. Spectroscopies are usually the more straightforward experimen-tal quantities to connect to the underlying theory because they often directly mea-sure a response function. In transport theory it is the elements of the appropriateconductivity tensor that are related to correlation functions [129]—current-currentfor the electrical conductivity. However the Hall resistivity (which is the quantitytypically measured in experiment) is not related to a single correlation. Rather


20

it is a ratio of conductivities through the inverse of the conductivity tensor. Oneis helped to some extent by the Onsager relation for the off diagonal elements ofthe conductivity (e.g. σxy = −σyx). Nevertheless the measured quantities can lookcomplex:

ρxy = − σxzσyz − σxyσzzσ2xzσyy + σ2

yzσxx + σ2xyσzz + σxxσyyσzz

tetragonal−−−−−−→symmetry

− σxyσ2xy + σ2

xx

. (7)

Fortunately for materials with high symmetry and in the weak field limit, thisratio can lead to cancellations between numerator and denominator. This can,in turn, give direct insight into physically measurable quantities, of which carrierconcentration in doped semiconductors is one elementary example. It is remarkablethat the heavy fermion systems may equally allow quite simple interpretations ofthe Hall effect as we will show later.

Although the Hall resistivity may not be directly related to a response function,it has been shown that the related quantity—the tangent of the Hall angle θH asdefined by eq. (3)—is [130]. Physically θH is the angle of deflection a free currentwould experience in a transverse magnetic field. Its role as a response functionmeans that the frequency dependent Hall angle can be integrated to provide a sumrule independent of the underlying microscopic physics [130]. Optical Hall anglemeasurements have been undertaken in the cuprates [131], but not yet in the heavyfermion metals.

3.2. Key results from Boltzmann theory

The most elementary theoretical treatment of the Hall effect is via a semi-classicalBoltzmann equation in the relaxation time approximation which is well describedin standard texts [132]. It assumes fermionic excitations with a Fermi surface deter-mined by a given dispersion relation and that the collision processes can be approx-imated by a decay time to equilibrium. In section 3.4 we consider the limitationsof this approach. Nevertheless, this is likely to be valid in the low temperaturelimit of the Fermi liquid state when impurity scattering dominates. The simpleresults of such an approach already illustrate a number of important features ofmagnetotransport in general.

Within the Jones-Zener expansion the conductivities are found order by order inmagnetic field [133], and are related to surface integrals around the Fermi surface.The conductivities are defined by

jα = σαβEβ + σ(1)αβγEβBγ + σ

(2)αβγδEβBγBδ + · · · , (8)

where

σαβ =

∫e2τ

∂f0

∂εvαvβ d

3k , (9)

σ(1)αβγ = εγδσ

∫e3τ2∂f

0

∂εvαvσM

−1βδ d3k , (10)

σ(2)αβγδ = εργνεµδσ

∫e4τ3∂f

0

∂εvαvρ

M−1σβM

−1µν +

vµ~

∂

∂kβ

(M−1σν

)d3k , (11)


21

εαβδ is the antisymmetric tensor, and the inverse effective mass is

M−1αβ =

1

~2

∂2ε

∂kα∂kβ. (12)

As can be seen, each conductivity is an integral over the Fermi surface (since theenergy derivative of the Fermi function, f0, is approximately a delta function).Furthermore, higher order dependence on the external magnetic field is relatedto a higher order derivative of the dispersion (εk) around the Fermi surface. Thissuggests that magnetotransport is, with each increasing order of field, a sensitiveprobe of the details of the Fermi surface.

These expressions reduce to the “pre-quantum” forms of Drude for the case ofa parabolic dispersion (σxx = ne2τ/m, σxy = ωcτσxx) so that the Hall coefficientRH = ρxy/B = 1/ne. However, they now account for the otherwise mysterious ob-servation that the Hall effect can change sign. This was first recognized by Peierlswho showed how the almost full band of electrons produces the same Hall effect asan almost empty band of positively charged carriers because of the sign of the cur-vature of the Fermi surface. In the case of a multiband system then conductivitiesarising from each band must be added together before the resultant conducivitytensor is inverted. This of course spoils the simple relationship between carrier den-sity and Hall constant even for a parabolic band. A peculiar case of note emergesfor compensated metals (where the Hall conductivity of the individual bands in,say, a two band system are equal and opposite and so cancel). Under these cir-cumstances the leading contribution to the measured Hall effect occurs at higherorder so ρxy ∼ B2 [80, 81]. The latter has been demonstrated to hold by high-fieldexperiments on the heavy fermion metal UPt3 [82]. This aspect will be importantfor the discussion of the Hall effect on CeM In5 systems, sections 6.3.2 and 6.3.3.

Some rather beautiful results have been proved by Ong within Boltzmann trans-port theory in the 2D limit which give geometric interpretation to magnetotrans-port quantities [25]. The Hall conductivity can be related to the flux through thearea swept out by the mean free path as it traced around the Fermi surface. Themagnetoresistance is related to the variance of the local Hall angle around theFermi surface [134].

The Boltzmann equation can also be solved in the limit of high magnetic field asan expansion in 1/B (though of course Landau level quantization means that thislimit is never strictly obtained). Under these conditions the Hall effect becomesinsensitive to Fermi surface shape and can be directly related to the volume ofthe Fermi surface. In this limit even in a multiband system one can use the Halleffect to determine the carrier concentration independent of band structure detailsRH = 1/e(

∑i ni). It is arguable as to whether this limit is ever physically relevant.

Finally, while the results quoted above relate to solutions obtained via expan-sion in magnetic field (either low field or high field), it is not necessary to solvethe Boltzmann equation by expansion. The conductivity can be obtained to all or-ders in the field at the expense of an additional surface integral using the so-calledChambers formula [135]. Such an approach is necessary when dealing with quanti-ties which vary rapidly around the Fermi surface as the Jones-Zener expansion canbreakdown. This will be important at a density wave transition for example (seelater).

These results from Boltzmann theory are often very useful as a starting pointto understand magnetotransport phenomena in the heavy fermion metals. In fact,sometimes they are all one can do. However, to make further progress we proceedto consider the foundations of this approach within Fermi liquid theory and thence


22

the physics beyond.

3.3. Hall effect within Fermi liquid theory

A more formal justification for the Boltzmann approach (and thereby revealing itslimitations) is obtained through the Kubo formula and its diagrammatic represen-tation. Direct methods of calculating typically involve treating the magnetic fieldperturbatively. Given we are looking for the current response to an electric andmagnetic field in the lowest order, the Hall conductance involves a three currentcorrelation. This issue of maintaining gauge invariance in these calculations requiresthat Ward identities be preserved [136]. In essence this means that the treatmentof vertex corrections must be consistent with the self-energy (i.e. that every pro-cess included in the self-energy has a corresponding vertex correction obtained byincluding a current vertex at any point within the self-energy diagram).

The simplest question concerns the recovery of the results of Boltzmann theoryin the relaxation time approximation. If one assumes non-interacting electrons anddilute impurities then these relaxation time approximation results above can beobtained if one assumes s-wave potential scattering [137]. The isotropic scatteringensures that the calculation is insensitive to vertex corrections. It also suggests thatin the case of strongly momentum dependent scattering which might be expectednear a QCP then reliance on the relaxation time approximation may be suspect.

Treating the problem of interacting fermions is considerably more difficult. How-ever, a remarkable simplification ensues if one is limited to systems where the lowenergy physics can be described within Fermi liquid theory [83]. In Landau Fermiliquid theory the low lying excitations of the interacting system are adiabaticallyconnected to a non-interacting Fermi gas (a pedagogical account being found inRefs. [138, 139]). Landau considered a neutral Fermi liquid in deriving the transportequation, but the extension of this equation to the charged case is straightforwardand dictated by minimal coupling. The derivation assumes that the Fermi liquid ischaracterized by a coarse-grained distribution of quasiparticles n(p, r, t) (i.e. the rdependence is slowly varying on the length scale of the Fermi wavelength). Underthese conditions the local quasiparticle energy can be considered to be the effectiveclassical Hamiltonian of the quasiparticle

ε(p, r) = ε0(p) +∑p′

fpp′δn(p′, r) , (13)

where the interaction is assumed to be local on the scale of spatial coarse graining.Minimal coupling introduces the electric and magnetic fields through the vectorpotential p → p − eA. The time evolution of the distribution is then determinedby the total time derivative

dn

dt=∂n

∂t+ ∇rn ·

∂r

∂t+ ∇pn ·

∂p

∂t= I , (14)

where I is the collision integral. Using Hamilton’s equation of motion and ex-pressing the result in terms of the deviation from local equilibrium, δn(p, r) =n(p, r)− n0(εp(r)), we find for steady state, translationally invariant solutions

eE ·∇pn0 + ev ×B ·∇pδn = I . (15)

This is exactly the result one would write down for simple Boltzmann transportyet it includes the electron-electron interaction within the Fermi liquid formalism.


23

Establishing that this result is rigorous within a diagrammatic approachwas achieved by Betbeder-Matibet and Nozieres [140] and by Khodas andFinkel’stein [141]. Thus the quasiparticle renormalization, Z, and other interac-tion corrections all cancel in the Hall coefficient at least to zeroth order in 1/τEF .Thus we conclude that if a Fermi liquid state emerges at low temperatures in theheavy fermion system, then transport can be described within a Boltzmann-likeformalism.

3.4. Quantum critical heavy fermion metals

All the results of the previous discussion require the existence of Fermi liquidlike quasiparticles. Understanding the Hall effect without quasiparticles is morechallenging [142]. Here we focus on what can be learnt from the Hall coefficientand higher order magnetotransport coefficients in the heavy fermion systems andwill argue that the most useful diagnosis is obtained in the T → 0 limit where aFermi liquid like picture should emerge. The heavy fermion metals hold particularinterest here since their inherently small energy scales mean that relatively weakperturbations like pressure or magnetic field are able to tune magnetic transitiontemperatures to absolute zero.

At zero temperature the QCP can be viewed a transition between Fermi liquidsbut the nature of that transition is a matter of very active interest. Two scenariosare emerging. On the one-hand, the nature of the QCP could be dictated by thefluctuations associated with a magnetic transition driven or tuned to zero temper-ature. This QCP might refer to as a conventional one—though as we will showthe theoretical description remains challenging in a number of physically relevantcases. In this case the Fermi liquids on either side of the quantum critical pointwould be both adiabatically connected to the same non-interacting Fermi gas (al-beit evolving continuously as the density wave gap opens at the transition point).

An alternative view is that the essence of the QCP in heavy fermion metalsis associated with the zero temperature breakdown of the Kondo effect. In thatscenario the magnetic instability plays a secondary role as a mechanism to quenchthe spin entropy of the local moments. In this scenario while the QCP woulddivide the zero temperature axis into two Fermi liquids, these two states wouldbe adiabatically connected to entirely different non-interacting systems. A centraltenant of this review is that the low temperature Hall data can play a key diagnosticrole in discriminating between these two scenarios.

3.4.1. “Conventional” Hertz-Moriya-Millis quantum criticality

We begin by considering the Hall effect at a “conventional” SDW QCP. Theeffective action for quantum criticality in a metal has been argued to be the socalled Hertz-Moriya-Millis action [4, 30, 31]

S =1

β

∫dDq

∑n

φ−q,−ωnE0

[r0 + ξ2

0q2 +|ωn|Γq

]φq,ωn

+u

∫dDrrr

∫ β

0dτ |φ(rrr, τ)|4 + · · · .

(16)In its simplest terms this action can be viewed as the quantum extension of theGinsburg-Landau free energy functional in the vicinity of a continuous phase tran-sition where the form of the expansion is deduced by symmetry and analyticity. Thefrequency sum of bosonic Matsubara frequencies is a consequence of the quantumnature of the transition (i.e. that the order parameter is not an eigenstate of theHamiltonian and so must be summed over in imaginary time). This action can beobtained from a random phase approximation (RPA)-like saddle-point treatment


24

of the interacting fermion problem [4]. The non-analytic term in ωn arises from thedamping induced by the electron fluid on the order parameter. The form of thedamping rate depends on the conservation laws applied to the order parameter:

Γq ∼ qz−2 =

Γ0 (z = 2) antiferromagnetic,vF q (z = 3) clean ferromagnet,Dq2 (z = 4) dirty ferromagnet .

(17)

The experimental consequences of this action have been well studied and so toohave been the conditions for its validity. These issues have been reviewed [5] but inbrief the low energy particle-hole excitations of the metallic state cannot generallybe safely integrated out to yield the action above [143]. The problem is particularlyacute for clean ferromagnetic instabilities where the soft particle-hole excitationsgenerate non-local terms in the effective action for the order parameter and wouldseem to generically drive the transition first order [144]. The case of two dimensionalantiferromagnets is also problematic from a theoretical standpoint [145].

Because of the theoretical questions concerning the precise nature of the magneticQCP in itinerant systems, for the purpose of this review we consider only theantiferromagnetic case and that at two levels. The first is the finite temperatureeffect of antiferromagnetic spin fluctuations on transport and the Hall effect. Thesecond is the zero temperature limit.

3.4.2. Spin-fluctuation theory

At a phenomenological level one can introduce a form of the spin-fluctuationsfirst introduced in the context of the high temperature cuprate materials [146]:

χsq(ω) =∑Q

χQ

1 + ξ2(q −Q)2 − iω/ωsf. (18)

Here ωsf and χQ scale with the square of the magnetic correlation length, ξ2, andQ are the Fourier modes of the incipient magnetic order. Transport is then dictatedby quasiparticle scattering from spin fluctuations. Two effects control the resultingtransport which can be considered within Boltzmann transport.

Firstly, the strong momentum dependence of the fluctuations strongly scattersparticles on the Fermi surface connected by the ordering wave vectors Q, whileleaving other parts of the Fermi surface relatively unscathed. The Fermi surfacedivides into ”hot-spots” where scattering is strong and “cold regions” where it isnot [147]. One issue within this scenario is the observation that the cold partsshould short-circuit the transport and so render the metal relatively insensitive tothe presence of the magnetic fluctuations [148]. Within such a picture a relaxationtime approximation has been invoked [149]. This produced a number of criticismsof the spin-fluctuation model as it applied to the cuprates. The origin of thesecriticisms is the observation made previously that magnetotransport is very sensi-tive to anisotropy around the Fermi surface and so a hot-spot–cold-region modelgenerates a large magnetoresistance [134, 150] (much larger than that seen in thecuprates).

The second effect is that the fluctuation performing the scattering is itself aparticle-hole excitation of the fluid and this modifies the current. This has beenemphasized by Kontani who recognized this effect within a tour-de-force diagram-matic approach to high order transport coefficients in a magnetic field (for a recentreview see Ref. [29]). In diagramatics this appears as a current vertex correction butits effect can be modelled within Boltzmann transport provided one does not make


25

the relaxation time approximation but considers the collision integral in detail.Taking both of these effects together, Kontani’s analysis of the spin fluctuation

model claims that one can obtain a strongly temperature dependent Hall coefficientin the presence of antiferromagnetic fluctuations: RH ∼ ξ2 ∼ 1/T . Moreover, higherorder terms in magnetotransport do not get anomalously large as they would inthe relaxation time approximation but adopt a temperature dependence associatedwith the Hall effect. One consequence is that the magnetoresistance ∆ρ/ρ0, whichin the relaxation time approximation would be expected to scale like ρ−2

0 (Kohler’srule, Ref. [151]), does not. Rather it would behave like ξ4/ρ2

0 ∼ T−4 leading to amodified Kohler’s rule ∆ρ/ρ0 ∼ (RH/ρ0)2.

The observations above are, at face value, very reminiscent of the behaviour of thenormal state of the high temperature cuprates superconductors. There, Anderson[27] and Ong [24] noticed that cot θH—which should, within the relaxation timeapproximation, be proportional to the scattering rate as measured in the resistivity(i.e. ∼ T )—here behaved with a distinct T 2 temperature dependence (cf. Fig. 28and related discussion in section 7.1.1; see also the discussion on results obtainedon CeM In5 materials, section 6.3.4). Crucial to their observation was the disorderdependence on the introduction of Zn impurities into the cuprates. They identifiedρ and cot θH as measuring entirely separate scattering rates because while bothhad very different temperature dependencies, they both also showed Matthiesen’srule type behavior as a function of impurity concentration: Impurity scatteringbehaved additively to both the scattering rates. In the context of the cuprates thisis a strong constraint on the underlying mechanism for the temperature dependenceof the Hall coefficient.

Adding disorder to a quantum critical antiferromagnetic metal was consideredby Rosch [152]. He showed that there is a strong interplay between even verysmall amounts of disorder and antiferromagnetic fluctuations. The disorder smearsout the hot-spots on the Fermi surface thereby considerably weakening the short-circuiting argument of Hlubina and Rice. Moreover, the expected temperature de-pendencies of the resistivity are also modified from those of the spin-fluctuationmodel. Rosch also showed that very small amounts of disorder are sufficient torender the Hall coefficient relatively weakly dependent on temperature [153]. Thisis somewhat at odds with the Kontani calculation which claims that the cot θHbehavior remains distinct from the resistivity albeit with a weaker than T 2 powerlaw.

Given that there remains some uncertainty about the precise tempera-ture/disorder dependence of the Hall coefficient when it is dominated by quantumcritical fluctuations, we argue that the T = 0 limit provides a more reliable domainfor the Hall effect’s interpretation. At temperatures low enough that the inelasticscattering is a relatively small fraction of the overall resistivity, one is dominatedby elastic impurity scattering. Under these circumstances a relaxation time ap-proximation is valid and one can use the magneto-transport data to characterizechanges in the Fermi surface.

Initial studies suggested that at a continuous density wave transition all transportquantities should vary smoothly [28, 48]. However, these considerations were basedon a weak field expansion. Crucially near a QCP there is an order of limits questionas to whether the field scale goes to zero before the density wave gap [50, 52]. Aweak field expansion is valid only if

Bτ <∆

ev2F

, (19)


26

where τ is the relaxation time and ∆ the density wave gap. Near enough to theQCP this expansion will breakdown for any finite field experiment in the orderedphase. The consequences are the Hall effect and resistivity develop non-analyticterms in field ωcτ . These arise whenever the Fermi surface crosses the density waveBrillouin zone so Bragg reflection causes the Fermi surface to develop a sharpcorner. Under these circumstances the Hall conductivity develops an additionalterm that goes like |ωcτ |2 and the longitudinal conductivity a term that behaveslike |ωcτ |. These terms are absent in the paramagnetic phase so this implies thatthere is a discontinuity in the Hall effect and in the resistivity of order |ωcτ |2and |ωcτ | respectively at the transition point. These discontinuities are roundedby magnetic breakdown effects and are suppressed by disorder. Nevertheless, theobservation of a magnetoresistance linear in magnetic field is often indicative of asharp feature (point of small radius of curvature) on the Fermi surface.

In summary, the theory of the Hall effect in a conventional density wave typeQCP is least ambiguous in the zero temperature limit. Here we expect to see rathersmooth evolution of the weak field Hall constant through the transition (unless thesystem is unusually clean when it may be possible to see discontinuities due tothe reconstruction of the Fermi surface). At finite temperature, Kontani predictsthat antiferromagnetic fluctuations can give rise to a temperature-dependent Hallcoefficient though Rosch argues that this temperature dependence is lost withrelatively small quantities of disorder.

3.5. Hall effect across Kondo breakdown quantum critical point

3.5.1. Quantum criticality in heavy fermion metals and jump of Hall coefficient

Theoretical studies of quantum phase transitions in heavy fermion metals departfrom the Kondo lattice Hamiltonian:

H =1

2

∑ij

Iij Si · Sj +∑kσ

εkc†kσckσ +

∑i

JK Si · sc,i . (20)

Here, a lattice of spin-1/2 local moments interact with each other with an exchangeinteraction Iij . To specify the typical strength of the exchange interaction, we useI to label the nearest-neighbor interaction, and we will focus on the case that it isantiferromagnetic. The model also contains a conduction-electron band, ckσ, witha band dispersion εk and bandwidth W . At each site i, the spin of the conduction

electrons, si,c = (1/2)c†iτττci, where τττ are the Pauli matrices, is coupled to a spin-1/2local moment, Si, via an antiferromagnetic Kondo exchange interaction JK .

When the Kondo coupling dominates over the RKKY interaction, the groundstate is a Kondo singlet (cf. section 2.5). The Kondo screening effect leads toKondo resonances, which are charge-e and spin-1/2 excitations. There is one suchKondo resonance per site, and these excitations induce a “large” Fermi surface[154–158]. Consider that the conduction electron band is filled with x electrons persite; for concreteness, we take 0 < x < 1. The conduction electron band and theKondo resonances will be hybridized, resulting in a count of 1+x electron per site.The Fermi surface would therefore have to expand to a size that encloses all these1 + x electrons. This defines the large Fermi surface.

Consider the conduction electron Green’s function:

Gc(k, ω) ≡ F.T.[− < Tτ ck,σ(τ)c†k,σ(0) >] , (21)


27

where the Fourier-transform (F.T.) is taken with respect to τ . This Green’s functionis related to a self-energy, Σ(k, ω), via the standard Dyson equation:

Gc(k, ω) =1

ω − εk − Σ(k, ω). (22)

In the heavy Fermi liquid state, Σ(k, ω) is non-analytic and contains a pole in theenergy space [154, 156, 157]:

Σ(k, ω) =(b∗)2

ω − ε∗f. (23)

Inserting eq. (23) into eq. (22), we end up with two poles in the Green’s function:

Gc(k, ω) =u2k

ω − E1,k+

v2k

ω − E2,k. (24)

Here,

E1,k = (1/2)[εk + ε∗f −

√(εk − ε∗f )2 + 4(b∗)2

],

E2,k = (1/2)[εk + ε∗f +

√(εk − ε∗f )2 + 4(b∗)2

](25)

describe the dispersion of the two heavy-fermion bands. These bands must accom-modate 1+x electrons, so the new Fermi energy has to lie in a relatively flat portionof the dispersion, leading to a small Fermi velocity and a large quasiparticle massm∗.

The self-energy contains only two parameters, the pole strength (i.e., theresidue), (b∗)2, and the pole location, ε∗f . eq. (23) describe only the coherent partwith the well-defined pole; an incoherent part will specify damping of the quasi-particle excitations.

In the opposite limit, when the RKKY interaction dominates over the Kondocoupling, the system is antiferromagnetically ordered. For any spatial dimensiond > 1, it follows from a renormalization group study [159] that the Kondo singletbreaks down. Correspondingly, there is no Kondo resonance, and the Fermi surfaceis “small”, i.e. does not incorporate the f -electrons. The conduction-electron self-energy in a large-N limit takes the form

Σ(k, ω) = aω − i(b|ω|d + cω2) sgnω (26)

From a comparison with eq. (23), we see that the strength of the pole in theconduction-electron self-energy has vanished. Inserting eq. (26) into eq. (22) givesrise to a Fermi surface that is entirely specified by the conduction electron disper-sion.

Quantum phase transition in the Kondo lattice arises by tuning the parameterδ ≡ T 0

K/I, where T 0K ≈ ρ−1

0 exp(−1/ρ0JK) with ρ0 being the density of states ofthe conduction electrons at the Fermi energy [89, 160] (cf. also Fig. 5). In recentyears, several theoretical approaches have been undertaken to study the quantumphase transition. The extended dynamical mean field theory (EDMFT) [161, 162]focuses on the destruction of the Kondo effect by the antiferromagnetic fluctua-tions, leading to two classes of solutions. In one, the Kondo breakdown occurs at


28

the onset of the antiferromagnetic order; this is referred to as local quantum crit-icality. In another class of solution, the Kondo breakdown takes place inside theantiferromagnetic order. The QCP separating the antiferromagnetic and param-agnetic states has the SDW form, but a Kondo breakdown gives rise to anotherquantum phase transition inside the antiferromagnetic part of the phase diagram.

The evolution of the Hall coefficient across the local QCP can be most clearlyseen in the zero-temperature limit. At δ > δc, the ground state is a Fermi liquid,and the conduction-electron self-energy has the form of eq. (23). The associatedquasiparticles, with the dispersion given by eq. (25), are located near the largeFermi surface. At δ < δc, the ground state is also a Fermi liquid, but now theconduction-electron self-energy, having the form of eq. (26), no longer has the poleand the quasiparticles are located near the small Fermi surface. As discussed earlierin this section, the quasiparticle residue for each Fermi liquid will cancel in its Hallcoefficient. Since the underlying non-interacting system for the two Fermi liquidscorresponds respectively the fermions with f -interactions itinerant or localized, theHall coefficient has different values in the two Fermi liquids [163]. In other words,the Hall coefficient at zero temperature jumps as the control parameter δ is tunedthrough δc, the QCP.

Another approach to the Kondo breakdown is based on a large-N formulation inconjunction with slave-particle representations of the spin operator. The fermionicrepresentation of the spin provides a means to access a spin liquid when the Kondoamplitude (b∗)2 is suppressed [164, 165], where fermionic spinons and the bosonic bfields are coupled to a gauge field. The Kondo destruction of b∗ = 0 corresponds toa metallic state (as opposed to a Bose-Einstein condensate) of the bosonic compo-nent. Calculation of the Hall coefficient and longitudinal resistivity in this approachshows a jump of both quantities at the Kondo-breakdown transition at zero tem-perature [166].

3.5.2. Crossover and scaling of the Hall coefficient at non-zero temperatures

The T -δ phase diagram for the Kondo-destruction local QCP was already illus-trated in Fig. 1(b). The energy scale E∗loc specifies a corresponding T ∗ line, whichseparates the phase diagram at low temperatures into two parts. To the right of theT ∗ line, the system flows (in the renormalization-group sense, as energy is lowered)towards a ground state with complete Kondo screening and therefore large Fermisurface; the TLFL scale describes the renormalized Fermi energy for this Fermiliquid state. To the left of the T ∗ line, however, the Kondo screening process is in-complete; in the ground state, the static Kondo screening is absent and the Fermisurface is small. Here, the Neel temperature, TN , marks the onset of the antiferro-magnetic order. In the zero-temperature limit, the end of the T ∗(δ) line, which isat δc, marks a genuine f -electron delocalization-localization phase transition.

Compared to the zero temperature case, the behavior of the Hall coefficientat non-zero temperatures across a Kondo-destruction local QCP is much moredifficult to analyze and can only be considered qualitatively. The single-electronGreen’s function, G(k, ω), on either side of the zero-temperature transition can bewritten as

G(k, ω) = Gcoh(k, ω) +Ginc(k, ω) . (27)

This decomposition is an immediate consequence of the fact that the phases sepa-rated by the QCP are Fermi liquids. The coherent part is given by

Gcoh(k, ω) =zk

ω − E(k) + iΓk(T )(28)


29

describing a quasiparticle, and Ginc(k, ω) is a background contribution. The quasi-particle residue, zk, is non-zero in either phase, but zk vanishes as the QCP isapproached: zk → 0 as δ → δc. At T = 0, the quasiparticle damping Γk vanishes atthe small Fermi-momenta kSF for δ < δc, and at the large Fermi-momenta kLF forδ > δc; at these respective Fermi-momenta, the quasiparticles become infinitely-sharp excitations at zero temperature. The coherent part of G(k, ω) is thereforethe diagnostic feature on either side of the transition, and it jumps at the QCP inaccordance with the sudden change of the Fermi surface. As already described, thisjump is manifested in the Hall measurement (see, e.g., section 5.1 where resultsobtained on YbRh2Si2 are presented), because the Hall coefficient is independentof the quasiparticle residue.

At non-zero temperatures, the quasiparticle relaxation rate at either kSF forδ < δc, or kLF no longer vanishes. In fact, inside the Fermi-liquid phase (witheither large or small Fermi surface), the temperature dependence of ΓkF

has to bequadratic in T . However, the Fermi surface remains well defined in these regimes.The change from one Fermi surface to the other is restricted to the intermedi-ate quantum critical regime. Because of the absence of a phase transition at anynon-zero temperature, the sharp reconstruction of the Fermi surface at T = 0 isturned into a Fermi-surface crossover across the T ∗(δ) line. This implies that therelaxation rate determines the parameter range corresponding the Fermi surfacechange. From the relation of the Hall coefficient to the Fermi surface we can asso-ciate the width of the Hall crossover with this broadening and consequently withthe relaxation rate Γ of the single-electron Green’s function.

Consider a general scaling form for the single-electron Green’s function at theFermi momentum in the quantum critical region:

G(kF , ω) =1

Tαg(kF ,

ω

T x

). (29)

The ω/T scaling at an interacting fixed point implies x = 1. The associated re-laxation rate, defined in the quantum relaxational regime (~ω kBT ) accordingto

Γ(kF, T ) ≡ [−i∂ lnG(kF, ω, T )/∂~ω]−1ω=0 , (30)

is linear in temperature: Γ(kF, T ) = cT , where c is a universal constant. Corre-spondingly, the crossover width is expected to be linear in temperature. This isindeed experimentally observed for YbRh2Si2, see Fig. 18 in section 5.1.

4. Experimental aspects of Hall effect measurements in metals

4.1. Measurement techniques

For the following considerations we rely on a discussion of eq. (1) for simplicity.Because of the typically large density of free charge carriers in a metal the Hallcoefficient RH is small. As an example, for aluminum RH is about −3.5 · 10−11

m3C−1 at room temperature [14]. Consequently, even for favorable experimentalconditions only a very small Hall voltage is generated. For Al and Ix = 100 mA,Bz = 1 T and d = 1 mm, the Hall voltage to be measured is as small as Vy = 3.5nV. The requirement to measure such small voltages calls for some preconditionsconcerning the experimental measurement setup as well as sample preparation. Wenote that these preconditions are specific for the investigation of metals. Semicon-ductors can exhibit values of RH in the order of 100 m3C−1 rendering Hall effect


30

Figure 8. Photograph of a single crystal YbRh2Si2 contacted for magnetotransport measurements. Thesample size is about 0.87× 0.6 mm2, with a thickness of about 65 µm.

measurements effortless. Therefore, all device applications based on the Hall effectuse a semiconductor rather than a metallic sample.

Obviously, one way of bringing up the Hall voltage independently of all measure-ment equipment is to thin the sample down. This is typically achieved by polishingor, in the final step, lapping. A single crystal YbRh2Si2 prepared in this way andinvestigated in section 5.1 is shown in Fig. 8. The sample thickness is about 65 µm.The contacts were spot-welded, with the left and right ones used as current leadsand the top and bottom ones for voltage measurements. Having two contacts alonga line parallel to the current Ix (lower sample side) allows for an additional mag-netoresistance measurement simultaneous with the Hall measurement. The samplewas embedded in varnish for good thermal contact and fixation. This sample wasthinned down along the crystallographic c-direction. According to Fig. 3 this im-plies Bz ‖ c. More generally spoken: Once the sample geometry is optimized thedirection with respect to the sample in which the magnetic field is to be applied isfixed. Consequently, a comparison of the Hall effect along different crystallographicdirections of a single crystal typically requires the preparation of one sample foreach envisioned direction of the applied magnetic field.

In principle, thin samples could also be prepared by thin film deposition tech-niques. Such films (e.g. of the manganites discussed in section 7.3) are well suitedfor Hall measurements if they are patterned. However, thin film deposition of heavyfermion metals has met limited success so far [167–170].

If the Hall measurements are to be conducted at cryogenic temperatures, self-heating effects of the sample due to the driving current Ix have to be prevented.This holds specifically true for the investigation of heavy fermion metals whichdevelop their most fascinating properties at lowest temperatures such that experi-ments are often run in the mK temperature range. For the experiments presentedin chapters 6 and 5, ac currents (frequency of 113 or 119 Hz) between 10− 100 µAwere applied. Comparing these currents to the example above demonstrates that,even though Ix should be adjusted as high as possible to improve the signal-to-noiseratio, the sample temperature needs to be monitored carefully.

A similar predicament holds for the magnitude of the magnetic field. From eq. (1),high fields appear preferable. However, it is typically the regime of linear response,i.e. the initial slope of ρH, one is interested in, see discussion of eq. (35) below andFig. 12 for an example. This constrain often limits the magnitude of Bz.

Any experimental setup to be used for Hall measurements on metals has to be op-timized for the measurement of small voltages. Besides careful cabling we fund theusage of low-temperature transformers [171] extremely helpful. These transformersallow for an impedance change (we typically amplify the voltage by a factor of100) very close to the sample and at low temperatures, i.e. at low noise level. Asa disadvantage, however, these transformers require magnetic field-free conditions


31

Figure 9. Sketch of a sample with contacts for van der Pauw measurements. An arbitrarily shaped sampleof uniform thickness is connected via four contacts on its circumference. The resulting Hall voltage isobtained by swapping the contacts for current and voltage (see text).

provided by their superconducting housing and hence, the magnet used for Hallmeasurements must have a compensated zone. If Hall effect and magnetoresistanceare to be measured simultaneously two transformers need to be implemented mak-ing space in a dilution refrigerator insert tight. Further amplification is providedby low-noise voltage [172] and, subsequently, lock-in amplifiers. The setup retainsphase control over the ac signals which is of utmost importance for crosschecks onthe measured Hall voltage.

A careful look at the sample contacts in Fig. 8 reveals an issue of far-reachingconsequences: The voltage contacts are usually not perfectly aligned perpendic-ularly with respect to current Ix (in contrast to patterned films). This mis-alignment causes a magnetoresistive component as part of the measured voltageVmeas(B) = Vy(B) + Vx(B). To separate the two components one can make useof the fact that the Hall effect is antisymmetric, Vy(B) = −Vy(−B), whereas themagnetoresistive part is symmetric, Vx(B) = Vx(−B). Experimentally, this re-quires to measure Vmeas(B) at positive and negative magnetic fields, doubling theeffort. The Hall voltage is then obtained by Vy(B) = 1

2 [Vmeas(B)− Vmeas(−B)]. Asan important way of crosschecking on our results we regularly compare Vx(B) =12 [Vmeas(B) + Vmeas(−B)] to the directly measured (e.g. via the two lower contactsin Fig. 8) magnetoresistive voltage; these two voltages are related by a simple factorwhich only depends on sample geometry but not on magnetic field.

For Hall measurements on samples of irregular shape one can use the so-calledvan der Pauw method [173] provided the sample is of homogeneous thickness andof platelet-like form, i.e., much thinner then wide. Four contacts are attached atarbitrary positions on the circumference of the sample, as illustrated in Fig. 9. Thevan der Pauw method is handy for tiny samples like those used in diamond anvilpressure cells. The Hall voltage is extracted from two subsequent measurementprotocols: First, the current is passed through one set of opposing contacts (e.g., 1and 3 in the figure) with the voltage measured on the transverse contacts (V24 in thefigure). In a second step, the former voltage contacts (2 and 4) are used to inject thecurrent and the voltage is measured on the former current contacts, giving again thetransverse voltage (V13 in the figure). The Hall voltage is calculated as VH = (V13 +V24)/2. An often applied procedure to separate magnetoresistivity contributionsarising from the arbitrary distribution of the contacts is via the determination ofthe antisymmetric component of the Hall voltage under field reversal.

The vectorial nature of the ingredients as described in section 4.2.1 calls fora tensor notation in defining the primary quantities of interest, specifically theresistivity %. Then, the Hall coefficient is given by RH = [%yx(B) − %xy(−B)]/2B.Though it is the Hall resistivity %xy which is always measured in an experiment, it is


32

common practice to report experimental data in the form of the Hall conductivityσxy using a full matrix inversion, i.e., σxy = %xy/[%

2xx + %2

xy]. Considering the factthat there is no preferred choice of notation in literature, this review uses both thesequantities interchangeably while describing prior data. Note that this requires theknowledge of both, %xx and %xy which is ideally measured simultaneously. Alongthe same lines we will use the notations H and B interchangeably to denote themagnetic field.

4.2. Advanced aspects of Hall effect measurements

4.2.1. Single-field Hall experiments

The traditional setup for Hall effect measurements was already sketched in Fig. 3.In the following we will refer to this as the single-field Hall setup with a singlemagnetic field B1 being applied, in an effort to distinguish it from the crossed-fieldHall setup discussed later, cf. section 4.2.2 and Fig. 10.

For a more general derivation of the Hall voltage, eq. (1), we consider the geom-etry shown in Fig. 3 and specify the boundary conditions as

jjj =

jx00

EEE =

ExEy0

BBB =

00B1

(31)

with the current density jjj, the electrical field EEE, and the magnetic field vector BBB.The conductivity tensor defined via Ohm’s law as

jjj = σEEE (32)

relates the current and the electrical field and incorporates dependencies of itselements on the magnetic field BBB. In the presence of one magnetic field in the z-direction only one may reduce the conductivity tensor to a 2× 2 form. By furthertaking into account the Onsager relations σxy(B1) = −σyx(B1) [174, 175] andassuming an isotropic material with σxx = σyy = σ, one derives a tensor with twoindependent components

σ =

(σ σxy−σxy σ

). (33)

We note that moderate anisotropies may be incorporated without altering themain result. In order to obtain an expression for the Hall coefficient we calculatethe resistivity tensor

ρ =

(ρ −ρxyρxy ρ

)=

1

σ2 + σ2xy

(σ −σxyσxy σ

). (34)

The Hall resistivity ρH is identical to the element ρxy as this element reflects therelation of the transverse electrical field to the longitudinal current correspondingto the definition of the Hall coefficient. We are now in the position to generalizeeq. (1) and obtain the linear-response Hall coefficient as the corresponding elementof the tensor on the right hand side of eq. (34)

RH(B1) = limB1→0

ρH

B1= lim

B1→0

1

B1

σxyσ2 + σ2

xy

≈ limB1→0

1

B1

σxyσ2

. (35)


33

single field crossed field

I

VH

B1

∂B1

I

VHB1

B2

(a) (b)

Figure 10. Hall effect setups to study field-induced QCPs. (a) Single-field setup with one magnetic fieldB1. (b) Crossed-field setup with the additional magnetic field B2.

The last approximation originates from the fact that the Hall conductivity in metalsis typically several orders of magnitude smaller as the normal conductivity, i.e.,σxy σ.

In accordance with eq. (35), the Hall coefficient is normally measured isother-mally as the initial linear slope of the Hall resistivity with respect to the magneticfield. For simple metals the Hall coefficient is a constant which implies that theHall resistivity is proportional to the magnetic field, ρH ∝ B1, cf. eq. (1). From thisequation it is also obvious that the Hall coefficient RH reflects the effective chargecarrier concentration which, for simple metals, does not depend on magnetic field.

However, for the investigation of more complex materials and/or phenomenasuch as quantum phase transitions the focus is often on non-linear effects. For thestudy of quantum criticality we are interested in metals which possibly change theirFermi surface topology and hence, also the effective charge carrier concentration.If this change is induced by a magnetic field, this will lead to a non-linearity in theHall resistivity. The overall magnetic field B1 applied will result in a tuning effect(cf. section 2.5) whereas the probing of the Fermi surface at this particular fieldstrength shows up as the response to infinitesimal fields ∂B1 (see Fig. 10(a)). It istherefore straightforward to define the differential Hall coefficient as

RH(B1) =∂ρH

∂B1

∣∣∣B1

(36)

which allows to monitor the evolution of the Hall response across a field-inducedQCP. Hence, the disentanglement of the two effects of the magnetic field might bedone in the subsequent analysis by numerical differentiation as we shall see below.

Based on linear response theory one may analyze the differential Hall coefficientin more detail. The orbital deflection of the charge carriers in an external magneticfield introduced by the Lorentz force leads to a term which represents a generalizeddefinition of the Hall coefficient:

RH(B1) =

[∂ρH(B1)

∂B1

]orb

(37)

In a simple metal the Lorentz force is just proportional to the external magneticfield giving rise to a constant RH as expected. This orbital contribution correspondsto the probe in our experiment, i.e., to the Hall response. When the Fermi surfacetopology changes, this is reflected in a change of the orbital contribution.

For the investigation of materials in which the ground state is field tuned, a sec-ond response to an external magnetic field arises, which is termed Zeeman term.Consequently, the differential Hall coefficient reflects both the orbital and the Zee-


34

man contribution:

RH =

[∂ρH(B1)

∂B1

]orb

+

[∂ρH(B1)

∂B1

]Zeemann

= RH(B1) +

[∂ρH(B1)

∂B1

]Zeemann

. (38)

We used eq. (37) to relate the first term to a linear-response of the system. TheZeeman term is not known to be related to any measurable linear-response quantity.Consequently, it is not straight forward to interpret results of the single-field Hallexperiment.

4.2.2. Crossed-field Hall experiments

The recent interest in heavy-fermion systems stems mainly from their modelcharacter in the investigation of QCPs. Although being phase transitions at zerotemperature quantum critical points lead to unusual properties up to surprisinglyhigh temperatures. This issue becomes particularly apparent for the cuprate su-perconductors for which quantum criticality is also discussed (see, e.g. [102] andsection 7.1.1). We shall see that the Hall effect turned out to be a sensitive tool toexplore the nature of a particular QCP.

Hall effect measurements provide currently the best probe to study the Fermisurface evolution at quantum critical points. This is mainly due to the experimentaldifficulties associated with other probes: ARPES is limited to higher temperatures(compared to magnetotransport experiments) whereas quantum oscillation mea-surements can only be conducted in a range of high magnetic fields and on veryclean samples.

In order to distinguish whether the Fermi surface evolves continuously or discon-tinuously one needs to have sufficient resolution of the control parameter used toaccess the quantum critical point. As the heavy-fermion systems involve magneticinteractions, magnetic field tuning is the obvious choice to achieve such high resolu-tion. This, however, means that the magnetic field has to cover two tasks, namely,tuning the material across its QCP and probing its Fermi surface by generatinga Hall voltage. Two different approaches can be pursued to disentangle these twoeffects.

Firstly, the conventional, single-field Hall effect setup as described in section 4.2.1might be employed. Here, the tuning effect stems from the overall applied magneticfield B1 whereas the probing of the Fermi surface at this particular field strengthshows up as the response to infinitesimal fields, see Fig. 10(a). The differential Hallcoefficient RH is given by eq. (36) and can be related to the Fermi surface via theKubo formalism [176]. Hence, the disentanglement of the two effects exerted by themagnetic field B1 is done in the subsequent analysis by numerical differentiation.

Secondly, by using two magnetic fields one can disentangle the two tasks alreadywithin the measurement. In order to do so, one magnetic field B1 is applied perpen-dicular to the current to generate the Hall response whereas the second magneticfield B2 is directed parallel to the current to tune the material, as sketched inFig. 10(b). We refer to this as crossed-field Hall effect setup. One precondition forthis disentanglement to successfully work out is, however, that the tuning effectarising from B1 has a negligible tuning effect to the material under investigationcompared to the tuning field B2. If the material exhibits a sufficiently high mag-netic anisotropy it can be utilized to ensure this condition. An influence of B2 onthe Hall effect is generally small as it is parallel to the current avoiding a Lorentzforce acting on the electrons. Consequently, tuning of the material is predominantly


35

achieved by B2 whereas the Hall effect stems from B1. The power of the crossed-field setup lies in the fact that the Hall effect represents the linear response of thesystem with respect to the Hall field B1: The Hall coefficient is extracted as theinitial slope of the Hall resistivity at a fixed tuning field B2,

RH(B2) = limB1→0

ρH(B1, B2)

B1. (39)

If the geometry between Hall field ∂B1, Hall voltage and injected current is keptidentical for both the single-field and the crossed-field configurations, the orbitalcontribution of the differential Hall coefficient RH(B1) and the linear-response Hallcoefficient RH(B2) respond to the evolution of the Fermi surface. Differences maybe ascribed to the Zeeman term or magnetic anisotropies as discussed below andin section 5.1.

We shall scrutinize the influence of B2 on the Hall response arising from B1

by analyzing the conductivity tensor for this setup. For the given geometry theboundary conditions may be framed as

jjj =

jx00

EEE =

ExEy0

BBB =

B2

0B1

(40)

Clearly, the conductivity tensor σ relating the current density and the electricalfield via eq. (32) will be a 3 × 3 matrix. Only the elements σxz = σzx = 0 canbe omitted as no magnetic field is applied in y-direction. Taking advantage of theOnsager relations [174, 175] one derives

σ =

σxx σxy 0−σxy σyy σyz

0 −σyz σzz

(41)

The resistivity tensor ρ may again be derived via inversion

ρ =1

σxxσyyσzz + σ2yzσxx + σ2

xyσzz

σyyσzz + σ2yz −σxyσzz 0

σxyσzz σzzσxx −σyzσxx0 σyzσxx σxxσyy + σ2

xy

(42)

from which the Hall coefficient can be read off as

RH = limB1→0

ρxyB1

= limB1→0

1

B1

σxy

(σxxσyy + σ2yzσxxσ

−1zz + σ2

xy). (43)

Rearranging eq. (43) allows a detailed comparison to the single-field case (B2 = 0):

RH = limB1→0

1

B1

σxyσxxσyy + σ2

xy

(1 +

σ2yz

σxxσzz

)−1

≈ 1

B1

σxyσ2

(1 +

σ2yz

σ2

)−1

. (44)

For the approximation in eq. (44) the same symmetry simplification as for the caseof the single-field Hall experiment, σxx ≈ σyy and the assumption of σxy σxxare used. Again, incorporating moderate anisotropies does not alter the results.In fact, it will become clear that one might utilize anisotropies to enable higherresolution in the crossed-field Hall experiment.


36

The leading contribution to RH in eq. (44) stems from the unity within thesum and matches the result of the single-field Hall experiment, eq. (35). Thiscontribution corresponds to the Hall constant in a gedanken setup in which thefield B2 does not generate any Lorentz force while fulfilling the role of tuning the

underlying state. The second part of the sum,σ2yz

σ2 , represents a correction to thiscase. In order to obtain an estimate for this correction we ignore anisotropies inthe material, i.e., we assume that the off-diagonal elements of the conductivitytensor are similar: σyz ≈ σxy. Thereby we derive that the correction to the idealcase without any direct effect of B2 on the Hall coefficient is given by

σ2yz

σxxσzz≈(ρH

ρ

)2

1 . (45)

We estimate this correction to be much smaller than 1 using again that for metalsthe off-diagonal elements of the conductivity tensor are typically several orders ofmagnitude smaller than the diagonal elements. Equation (45) denotes the squareof the tangent of the Hall angle (tan θH)2 which is typically of the order of 10−4

for metals at low temperatures and small fields. Consequently, this correction maybe neglected.

These considerations of the transport theory emphasize that the single-field Hallsetup allows to study the Hall effect evolution across a QCP by means of the differ-ential Hall coefficient which is composed of an orbital response and a Zeeman term.By contrast, the crossed-field Hall experiment exclusively yields the linear-responseHall coefficient associated with the orbital effect. Consequently, the crossed-fieldHall setup appears to be superior. However, we shall see that the crossed-field setupis experimentally more demanding. The consistency of the two setups was testedby investigating a non-magnetic metal [177], see section 4.2.4.

4.2.3. Realization of crossed-field Hall experiments

For many investigations access to lowest possible temperatures is desired oreven required. Hence, Hall effect measurements are often performed in a 3He/4He-dilution refrigerator. For the case of the crossed-field Hall setup a vector magnetcomprising at least to components is favorable. Such a setup is sketched in Fig. 11.Here, the Hall field B1 is generated by a solenoid oriented vertically whereas thetuning field B2 is generated by a split pair oriented horizontally. Superconductingmagnets in persistent mode allow low noise measurements of the Hall signal.

This setup is clearly superior to an earlier realization of crossed fields [176] inwhich a miniature superconducting coil (providing B2) was mounted inside a dilu-tion refrigerator equipped with a conventional solenoidal superconducting magnet(used to apply B1). In the latter case, problems arose due to the limited space (lim-iting the magnitude of B2), possible quenching of the miniature coil and a difficultalignment of the coils and the sample. We note that the tuning field B2 should beadjustable.

The Hall voltage is measured as a function of B1 at several field strengths ofB2. This allows to extract the Hall coefficient RH(B2) as the initial linear slopeof the Hall resistivity ρH with respect to B1 at fixed B2, cf. eq. (39). Again, theHall voltage is to be measured at negative and positive fields B1 to enable theanalysis of the antisymmetric part (cf. section 4.1). As we shall see below it isfavorable to measure also at positive and negative B2 to correct for misalignments.Consequently, the Hall voltage has to be scanned on a close mesh of B1 and B2

(and T in order to allow an extrapolation to T → 0). The effort is partially reducedby the fact that it is sufficient to restrict the range of B1 to small fields as we are


37

Figure 11. Experimental realization of the crossed-field Hall setup. A 3He/4He-dilution refrigerator iscombined with a vector magnet comprising a solenoid generating a field in the vertical direction and asplit pair generating a horizontal magnetic field, parallel to the current injected into the sample (notshown).

only interested in the initial slope of ρH(B1)|B2. The second magnetic field B2,

however, gives rise to a large increase in measurement time.The experimental procedure is illustrated in Fig. 12 for the measurements on

YbRh2Si2 at one particular temperature (T = 65 mK). Here, the Hall resistivityvs. B1 clearly exhibits a linear field dependence. Importantly, however, the slope ofthese curves, i.e. RH, is different for different tuning fields B2. The slope of ρH(B1)was extracted from least square fits to determine the dependence of RH(B2). Thisis to be done for several temperatures in order to enable an extrapolation to zerotemperature.

For the conduction of the crossed-field Hall experiments it is necessary to keepthe tasks of the magnetic fields well separated. As already pointed out, the tuningeffect of the Hall field B1 should be small compared to the tuning effect of thetuning field itself. This may be assured by applying small field B1 only, but thisalso decreases the Hall signal and hence, the sensitivity. Note that also for thesingle-field Hall setup one is faced with this issue. For a material with a smallcritical field it will be hard to resolve non-linear effects as the Hall signal will besmall around the critical field. Large critical fields on the other hand will lead to anincrease of the Zeeman response in the single-field Hall setup and might easily bebeyond the field scale accessible with vector magnets needed for the crossed-fieldHall setup.

One may partially overcome the above described issues by taking advantage ofanisotropies in the materials response to external fields. For the case of YbRh2Si2which is discussed in more detail in section 5.1, the magnetic anisotropy is such thata field along the crystallographic c direction has to be 11 times larger comparedto a field within the crystallographic ab plane to generate the same tuning effecton the ground state of the material. Clearly, this favors an application of the Hallfield B1 along the c direction and, thus, in the case of the crossed-field experiment,the tuning field B2 within the easy ab plane. This allows to use higher fields B1

inducing larger Hall voltages without generating a substantial tuning effect. In


38

0.0 0.2 0.40.0

0.1

0.2YbRh2Si2 sample 2T = 65 mK

B 2 = 0 mT

200 mT

58 mT

B1 (T)

H (1

0-10 Ω

m)

Figure 12. Exemplary Hall resistivity isotherms for one YbRh2Si2 sample. Solid lines mark linear fits tothe data with the slope of these lines corresponding to the linear-response Hall coefficient RH.

the case of YbRh2Si2 it turned out that fields B1 . 0.4 T could be used withoutdisturbing the tuning effect generated by B2. In fact, the Hall resistance remainedlinear also in proximity to the critical value of the tuning field where the Hallcoefficient RH(B2), and thus the slope of the Hall resistivity considerably changes,as can be seen from Fig. 12.

One of the major issues of the experimental setup as depicted in Fig. 11 is thealignment of the magnetic fields with respect to the sample. This appears to bean issue for both the single-field and crossed-field Hall setup as misalignments canlead to unwanted effects. For the single-field Hall setup it is important to havea precise alignment of B1 perpendicular to the current and to the Hall voltagecontacts. Any component out of this plane will generate a tuning effect withoutinducing a Hall response. This effect will be amplified by a magnetic anisotropy.However, one may take advantage of the magnetic anisotropy to align the samplewith respect to the magnetic field (or vice versa). A proper orientation may beassured by monitoring characteristic features related to the materials response tomagnetic fields as a function of angle between sample and magnetic field. For theexperiments on tetragonal YbRh2Si2 a precision of better than 0.5 in alignmentof B1 parallel to the magnetic hard c direction was achieved by maximizing thecritical field while rotating the sample with respect to the field direction [178].

For the crossed-field Hall setup a misalignment might lead to a mixing of theintended effects of the two magnetic fields. Any component of B1 within the plane ofthe current and Hall voltage contacts acts as a tuning field, and a component of B2

perpendicular to current and Hall voltage contacts contributes to the Hall field B1.Even small misalignments of the order of 1 may change the results dramaticallyin the presence of a magnetic anisotropy (as in case of YbRh2Si2). In fact, a tiltingof the sample may lead to an antisymmetric part in the longitudinal voltage as afunction of B1 if B2 is finite. This is in contrast to the fact that the longitudinalresistivity, i.e., the magnetoresistance ρxx(B) is a symmetric function of magneticfield. This effect arises solely from the non-trivial combination of the two magneticfields. For the measurement of the Hall coefficient in the crossed-field setup thisis crucial because a longitudinal component will be present on the Hall voltagecontacts as well. This component is usually separated by taking the antisymmetricpart of the measured voltage to derive the Hall resistivity. This, however, fails ifthe longitudinal component already contains an antisymmetric part due to samplemisalignment. Such an antisymmetric part would consequently add to the Hallcoefficient. As a sample is manually aligned to a precision of the order of severaldegree only one might correct such a misalignment by tilting back the data. For the


39

0 1 2 3 43.0

3.5

4.0

4.5

5.0

0.0 0.2 0.4 0.6 0.8 1.0

3.5

4.0

4.5

5.0

5.5~

~

RH(B

1)

single-field 5

RH (

10-1

0 m3 /C

)

B1(T)

RH(B

2)

crossed-field T (K)

0.02 0.065 0.3 1

LuRh2Si

2

RH (

10-1

0 m3 /C

)

B2(T)

Figure 13. Comparison of crossed-field and single-field Hall coefficient of LuRh2Si2Crossed-field resultsRH(B2) are plotted on the lower and left axis, whereas single-field results RH(B1) are plotted on the upperand right axis. Solid line emphasizes the linear evolution of both at small fields. The field axis of the single-field data is scaled by a factor of 4 with respect to the crossed-field data accounting for the anisotropyobserved in the field dependence of the Hall coefficient. In addition, the ordinate of the crossed-field datais shifted by 0.475× 10−10 m3/C compensating the reduction of RH in the crossed-field experiment (seetext). Reproduced with permission from S. Friedemann et al., Physica Status Solidi B 247 (2010) 723[177]. Copyright c© 2010 WILEY-VCH.

experiments on YbRh2Si2, the uncertainty could be reduced to a tilting in the planespanned by B1 and B2. Consequently, it was possible to analytically rotate backthe Hall voltage VH(B1, B2). Therefore, both the longitudinal voltage Vx and theHall voltage Vy were measured simultaneously as a function of B1 and B2, with B2

ranging from positive to negative values. The analysis of the longitudinal voltagerevealed an antisymmetric component which could be omitted when rotating thedata by a small angle. The Hall voltage was then corrected by the same angle priorto the calculation of the Hall resistivity and the Hall coefficient [178].

4.2.4. Comparison of single and crossed-field Hall experiments

The consistency of the single-field and crossed-field Hall setup was checked for anon-magnetic metal, LuRh2Si2 [177]. In these two measurements a linear decreaseof the differential and linear-response Hall coefficient was revealed, respectively.The comparison of the results is illustrated in Fig. 13. First, the single-field Hallresults (open symbols) display a linear decrease as the magnetic field is increased upto B1 = 1 T. Above this field a saturation gradually sets in. Second, in the crossed-field Hall experiment a linear decrease of RH(B2) is observed over the complete fieldrange studied, i.e., up to B2 = 4 T. The fact that the change of the Hall coefficientis linear in both experiments suggests a Zeeman splitting of the Fermi surfacesas the underlying mechanism. The different field range over which the linearityoccurs might be related to the anisotropy of the material: B1 and B2 are appliedalong different crystallographic directions. In fact, the data collapse if the crossed-field Hall data are scaled by B2/B1 = 4 (note that the anisotropy in LuRh2Si2 isnot only smaller than in YbRh2Si2 but also reversed). The different temperaturesof the crossed-field and single-field Hall data do not affect these considerationsas both crossed-field and single-field Hall results do not exhibit any temperaturedependence below 20 K.

In the crossed-field experiment, the Hall field B1 also influences the tuning ofthe system due to the unfavourable anisotropy. Therefore, a disentanglement of thetwo effects of the magnetic field is incomplete in contrast to the case of YbRh2Si2,as we shall see in section 5. In fact, the results on LuRh2Si2 in the crossed-fieldconfiguration demonstrate that the tuning effect can arise from both the dedicated


40

tuning field B2 as well as from the Hall field B1. Here, the Hall coefficient isdeduced as the positive linear slope of the Hall resistivity over a finite field range.In this field range, however, the Hall resistivity is sublinear as best seen from thedecrease of the differential Hall coefficient (see Fig. 13). Consequently, the slopeof the linear fits and thus the linear-slope Hall coefficient reflects an average ofthe differential Hall coefficient over the field range considered for the linear fits.This leads to a reduction of the linear-slope Hall coefficient with respect to thedifferential Hall coefficient which is reflected in Fig. 13 as a negative offset of thecrossed-field Hall results. This confirms that for the case of LuRh2Si2 in the crossed-field Hall experiment tuning stems from both B2 and B1. The tuning contributionof B1 can be minimized by either going to smaller fields B1 which makes the Hallsignal harder to detect, or by utilizing magnetic anisotropies such as described forYbRh2Si2.

This section reveals the experimental difficulties associated with the two setups.Although one should not underestimate the issues of the single-field Hall setup,the crossed-field experiment is much more involved. In case of the single-field setup(which is very much established for the investigations of quantum critical phenom-ena) one should consider the effort of low-temperature measurements and preciseorientation. For the crossed-field Hall setup the effort is extended by the additionalmagnet needed and the time consumption to scan the second magnetic field. How-ever, due to the difficulties with the interpretation of the single-field Hall results itis necessary to combine them with crossed-field Hall measurements. As we shall seein section 5.1, the crossed-field Hall effect measurements seem to be the superiorprobe to reveal a Fermi surface reconstruction at a QCP.

5. Hall effect and Kondo-breakdown quantum criticality

5.1. Hall effect evolution at the QCP in YbRh2Si2

The heavy-fermion material which was very extensively studied by Hall effect mea-surements across its QCP is YbRh2Si2. We discuss the results on this material andsubsequently compare to other heavy-fermion materials. Hall effect measurementson YbRh2Si2 provide currently the best evidence for a Fermi surface reconstruc-tion at a QCP. In fact, these Hall effect measurements allowed also to extractinformation on the quantum critical scaling behavior as we shall see below.

YbRh2Si2 is a prototypical material for the investigation of quantum criticalphenomena. The heavy-fermion character is evident from various transport andthermodynamic properties. The resistivity ρ exhibits a logarithmic temperaturedependence above 100 K associated with the single ion Kondo scattering. Below100 K, ρ(T ) [179, 180] as well as the thermopower [180] decrease as the tempera-ture is reduced reflecting the onset of coherent Kondo scattering and the formationof composite quasiparticles. These effects, however, involve all crystalline electricfield (CEF) levels [181]. Upon cooling, the 4f electrons increasingly “condense”into their CEF-derived Kramers doublet ground state allowing the formation ofa Kondo lattice below around 30 K [179, 180]. These effects of combined CEFand Kondo interactions have recently been demonstrated by scanning tunnelingspectroscopy on YbRh2Si2 [182]. Only at very low temperatures of TN = 70 mKorders YbRh2Si2 antiferromagnetically [179]. The Neel temperature TN is continu-ously suppressed by a small magnetic field leading to the QCP. This field-inducedQCP is accessed at a critical field of either B1c = 660 mT if applied along thecrystallographic c direction, or B2c = 60 mT if applied perpendicular to the c axis.This magnetic anisotropy is used to the advantage of separating the two magnetic


41

field tasks in the crossed-field experiment (cf. 4.2.3). The phase transition appearsto remain continuous down to at least 15 mK [183] indicating gapless quantumfluctuations. Right at the critical field pronounced non-Fermi liquid behavior isobserved over three decades in temperature down to the lowest temperatures ac-cessible [184]. This non-Fermi liquid behavior is best explained to arise from thequantum fluctuations within the QCP picture which assumes a continuous phasetransition at zero temperature. The non-Fermi liquid behavior comprises a loga-rithmic divergence of the Sommerfeld coefficient of the electronic specific heat, i.e.γ = Cel/T ∝ log(T/T0), and a linear temperature dependence of the resistivity[179, 184]. At fields larger than the critical field a Fermi liquid ground state is re-covered. Here, the resistivity follows a quadratic temperature dependence and theSommerfeld coefficient saturates. The characteristics of this Fermi liquid phase arestrongly enhanced as typical for a heavy fermion material, the quadratic coefficientof the resistivity and the Sommerfeld coefficient obey values up to a factor 1000larger than in non-correlated metals [184]. This reflects an extremely large effectivemass of the composite quasiparticles.

Indications of an unconventional type of quantum criticality arose from measure-ments of the Gruneisen ratio [185]. The temperature dependence of the Gruneisenratio strongly underpins the presence of a QCP as it obeys a power-law diver-gence, i.e., Γ ∝ T−x. The observed exponent of x = 0.7 is, however, in contrastto the predictions of the conventional SDW QCP (x = 1). This fact along withthe outcome of B/T scaling of both the specific heat and the electrical resistiv-ity [7] suggests that YbRh2Si2 is an ideal candidate for unconventional quantumcriticality. This motivated extensive Hall effect studies on YbRh2Si2 [53, 176]. Asthe Hall coefficient is a rather complex quantity in its relation to the Fermi sur-face (cf. section 2.4) we should first note some peculiarities of the Hall effect inYbRh2Si2. First, the anomalous Hall effect is shown to be dominant at high tem-peratures but is negligible at lowest temperatures [186]. In addition, we shall seethat the anomalous Hall contributions can not account for the signatures relatedto quantum criticality. Second, we note that the low temperature Hall coefficientof YbRh2Si2 is found to exhibit sample dependencies. For the almost compensatedtwo-band material YbRh2Si2 these sample dependencies where found to arise fromslight changes of the relative scattering rates of the two dominating bands [187].These differences in the relative scattering rates appear to originate from tiny vari-ations in the composition because they only affect samples from different batches.A systematic study of different samples revealed the sample dependencies to beassociated with a background contribution whereas the contributions related tothe quantum criticality are robust [53].

The Hall effect on YbRh2Si2 was studied in both the single-field and the crossed-field geometry (explained in section 4.2). These studies are further corroboratedby longitudinal magnetoresistance measurements (see Ref. [49]). In the single-fieldHall experiment, a single magnetic field, B1, applied along the c axis was used toboth tune and probe the sample. For the crossed-field Hall experiment, the tuningwas carried out with an additional magnetic field B2 within the crystallographicab plane while a small field B1 was utilized to probe the system. The scales ofthe tuning fields, i.e., B1 ‖ c and B2 ⊥ c, of the two experiments differ by themagnetic anisotropy of the material. This is quantified by the ratio of the criticalfields B2c/B1c = 1/11. We shall use this ratio to convert the single-field Hallresults to the equivalent crossed-field scale. As detailed in section 4.2.2 the latterorientation allows to probe the system at larger fields B1 thus increasing the Hallsignal. Fig. 12 illustrates that in an extended range of low fields up to B1 = 0.4 T,the Hall resistivity is linear with respect to B1 even close to the critical field B2c


42

0.0 0.1 0.2 0.3 0.4

0.0

0.2

0.4

0.6

0.8

0.0 0.1 0.2 0.3

0

2

4

6

8

YbRh2Si

2 sample 2

T (mK) 100 65 45 20

RH (

10-1

0 m3 /C

)

B2 (T)

-∂R

H/∂

B2

(10-1

0 m3 C

-1T

-1)

B2 (T)

FWHM

B2c

Figure 14. Crossed-field Hall effect results: Linear-response Hall coefficient measured in the crossed-fieldHall experiment on YbRh2Si2. The field dependence of the isothermal linear-slope Hall coefficient is derivedfrom linear fits to the Hall resistivity ρH(B1)|B2,T

as detailed in section 4.2.2 (Fig. 12). Solid lines are

best fits of eq. (47) to the data extending up to 2 T. The inset shows the derivative −dRH/dB2 of both thedata (symbols) and the fits (solid lines). Vertical arrow indicates the critical field B2c. Horizontal arrowsspecify the FWHM of the peak for one curve. Reproduced from S. Friedemann et al., Proceedings of theNational Academy of Science USA 107 (2010) 14547 [53].

which, in consequence, allows to reliably extract the linear-response Hall coefficientas a function of B2. The longitudinal magnetoresistance as shown in detail inRef. [49] was measured with field perpendicular to the c-axis simultaneous to thecrossed-field Hall effect and is therefore associated with the equivalent field scaleB2.

Figs. 14 and 15 display selected isotherms of the linear-response Hall coefficientRH(B2) and the differential Hall coefficient RH(B1) deduced in the crossed-fieldand single-field Hall experiment, respectively. For plots of the magnetoresistancedata we refer the reader to Ref. [49]. All data sets exhibit similar behavior compris-ing two main features. First, at small, increasing fields the Hall coefficient and themagnetoresistance decreases in a pronounced, crossover-like fashion. Second, at ele-vated fields and low temperatures, the crossover of the Hall coefficient is followed upby a linear increase. With decreasing temperature the crossover becomes sharper infield, decreases in height, and is shifted to lower fields in both experiments. In fact,it approaches the critical field in all cases, i.e., B1c and B2c for the single-field andcrossed-field Hall and magnetoresistance experiment, respectively. Consequently,the crossover is related to the quantum critical behavior. With the crossover beingrestricted to small fields the linear behavior is revealed over an increasing fieldrange as the temperature is lowered. This suggests that the linear behavior repre-sents a background contribution to which the crossover is superposed. The linearityof the background contribution suggests that it originates from Zeeman splitting.Moreover, it is subject to the above mentioned sample dependencies. The mainfeatures of the curves, i.e. the crossover, by contrast, are not affected by the sam-ple dependencies. The decomposition of the Hall coefficient into a critical and abackground contribution is further illustrated in the inset of Fig. 14. Here, thederivative −∂RH/∂B2 is shown. The crossover is reflected by a peak around thecritical field (indicated by a vertical arrow) whereas the background contributionappears as a non-zero offset. Below, we shall examine the characteristics of thecrossover in the Hall coefficient in more detail and relate it to the quantum critical


43

0 1 2 3 4-1

0

1

2

∂ρH/∂

B1 (

10-1

0 m3 C

-1)

YbRh2Si

2

sample 2

T (mK) 300 200 120 50 20

B1(T)

Ra

H

Figure 15. Single-field Hall effect results: Isotherms of the numerically derived differential Hall coefficientRH(B1) of YbRh2Si2. Solid lines mark best fits of eq. (47) to the data. The vertical bar denotes the size ofthe constant offset originating from an anomalous contribution estimated from eq. (46) via Ra

H = ∂ρaH/∂B1.Reproduced with permission from S. Friedemann et al., Journal of Physics: Condensed Matter 23 (2011)094216 [49]. Copyright (2011) IOP Publishing Ltd.

contribution.An anomalous contribution to the Hall effect in YbRh2Si2 appears to be neg-

ligible at lowest temperatures. The skew scattering (cf. section 2.3.1) typicallyimportant in heavy fermion systems at high temperatures was found to lead topronounced signatures in RH(T ) around 100 K. At lowest temperatures, however,this contribution is largely suppressed. The fact that the anomalous contributionto the Hall resistivity

ρaH(B1) = Cρ(B1)µ0M(B1) (46)

is essentially linear in field implies that it adds a small constant offset to the Hallcoefficient. Using the constant C determined at higher temperatures [186] leads toa value of ρa

H = 0.07 × 10−10 m3/C which is tiny compared to the large variationof the differential Hall coefficient as illustrated by the vertical bar in Fig. 15. Moreimportantly, the added offset to the Hall coefficient does not affect the analysisof the critical contribution. Particularly, the width, height and position of thecrossover are independent of any offset.

For the detailed analysis the empirical function

RH(B2) = R∞H +mB2 −R∞H −R0

H

1 + (B2/B0)p(47)

was fitted to the data. This function simulates a crossover from the zero-fieldvalue R0

H to the high-field value R∞H superposed to a linear background mB2. Theposition of the crossover is given by B0 and the sharpness is determined by p whichis restricted to positive values. The function describes the data very well as can beseen from Figs. 14 and 15. Analogous fits to the differential Hall coefficient RH(B1)measured in the single-field experiment yielded the zero-field and high-field valuesR∞H and R0

H, respectively. Equation (47) allows to deduce the characteristics of thecrossover, i.e., the quantum critical contribution. In the following we shall focus


44

0.0 0.1 0.2 0.3 0.40.0

0.1

0.2

0.3

crossed- single- magneto- field field resistance

YbRh2Si

2 R

H(B

2) R

H(B

1) ρ(B

2)

sample 1 sample 2

TLFL

TN

T (

K)

B2 (T)

FWHM(T)

~

Figure 16. Linkage of the Hall crossover to the QCP in YbRh2Si2. The crossover field B0 was extractedfrom fits of eq. (47) to the crossed-field RH(B2) (solid symbols) and single-field RH(B1) (open symbols)Hall results as shown in Figs. 14 and 15 as well as to magnetoresistance ρ(B2) (Ref. [49]). Results on twosamples with different residual resistivities match within experimental accuracy. The values deduced fromthe single-field Hall experiment are scaled by 1/11 to account for the magnetic anisotropy of the material(see text). Red horizontal bars represent the FWHM at selected temperatures. Error bars are omitted toavoid confusion with the FWHM bars; with exception of the data at 0.3 K and the single-field result onsample 2 at 0.19 K standard deviations are smaller than the symbol size. Dotted and dashed lines markthe boundary of the magnetically ordered phase and Fermi-liquid regime, respectively, as taken from Ref.[184]. Reproduced from S. Friedemann et al., Proceedings of the National Academy of Science USA 107(2010) 14547 [53].

on a complete characterization of the crossover with respect to its position, widthand height.

Firstly, the position of the crossover is tracked in the phase diagram shown inFig. 16 for two samples of very different values of the residual resistivity whichspan the whole range of sample dependencies observed for the low-T Hall coeffi-cient (cf. Ref. [187] with identical nomenclature). The crossover field is found tomatch for both samples and for three different experiments, i.e., for crossed-fieldand single-field Hall effect measurements and for the analogous crossover in themagnetoresistivity. As the temperature is lowered B0 shifts to lower fields and con-verges to 60 mT, i.e., it extrapolates to the QCP in the limit of zero temperature.This reveals the linkage of the crossover in the Hall coefficient (and magnetoresis-tance) to the quantum criticality.

Secondly, the height of the crossover is characterized by the limiting parametersR0

H and R∞H . Their temperature dependencies are plotted in Fig. 17 on a quadratictemperature scale which allows to identify important temperature dependencies aswe shall see in the following. The absolute values of R0

H and R∞H differ for the twosamples, in accordance with the sample dependencies that are associated with thebackground contribution to RH. The quantum critical contribution is characterizedby the difference R0

H−R∞H . This difference decreases as the temperature is loweredwhich can be seen from the RH(B2) curves in Fig. 14. In order to draw conclusionson the evolution of the Fermi surface at the QCP we need to deduce this differencein the limit T → 0. Therefore, we utilize the temperature dependencies found forthe initial-slope Hall coefficient. The zero-field value R0

H extracted from the fitsmatches very well with the measured initial-slope Hall coefficient in zero tuningfield. We can therefore use the quadratic temperature dependence found for theinitial-slope Hall coefficient below TN to reliably extrapolate to T → 0 (cf. solidlines in Fig. 17) [49]. The temperature dependence of the high-field value R∞H is lesspronounced and may well be approximated with a quadratic form as well. Impor-


45

0 2 4 6 8 10

0.0

0.5

1.0

2

3

0 2 4 6 8 10

-1

0

1

2

3 sample panel 1 / 2 (a) (b)

/ RH RH

/ R0H R

0H

/ R∞

H R∞

H ~

~

TN

(a)

sample 1

sample 2

YbRh2Si2

RH (1

0-10 m

3 /C)

T 2 (10-3 K2)

TN

YbRh2Si2

T 2 (10-3 K2)

(b)

RH (1

0-10 m

3 /C)

TN

TN

Figure 17. Limiting parameters of the Hall crossover in YbRh2Si2. The initial-slope Hall coefficient (opensymbols) is plotted against a quadratic temperature scale for two selected samples. Solid lines mark bestfits of a quadratic form RH(T ) = RH(T = 0) + A′T 2 to the initial-slope Hall coefficient. The zero-field(R0

H) and the high-field (R∞H ) values extracted from fits of eq. (47) to RH(B2) are included in (a). The

corresponding values (R0H and R∞H ) extracted from the single-field Hall results (Fig. 15) are displayed in

(b). Arrows mark the Neel temperature. Reproduced with permission from S. Friedemann et al., Journalof Physics: Condensed Matter 23 (2011) 094216 [49]. Copyright (2011) IOP Publishing Ltd.

tantly, the extrapolation of R0H(T ) and R∞H clearly indicates that their difference

persists in the extrapolation T → 0, i.e., down to the QCP. This is not only truefor R0

H − R∞H of both samples but also for the difference R0H − R∞H of the limiting

values extracted from the single-field Hall experiment displayed in Fig. 17(b) aswell as for the corresponding values of the crossover in magnetoresistance [49]. Insummary, we can conclude that a change in the Hall coefficient persists at the QCPand is robust against sample dependencies.

The absolute value of the crossover height, R0H − R∞H , is difficult to evaluate

because multiple bands almost compensate each other at EF (despite the fact thatthe valence of ∼2.9 indicates YbRh2Si2 to be a hole system [188]) and these bandsare influenced by sample dependencies [187]. Yet, the drop of RH at B0 is consistentwith a change from two hole-like Fermi-surface sheets dominating at B < B0 toone hole/one electron sheet for B > B0 due to the hybridization between 4f andconduction electrons in the latter case.

Finally, we examine the width of the Hall crossover. This is accomplished bydetermining the full-width at half-maximum (FWHM) of the derivative of thecurves fitted to RH(B2), see inset of Fig. 14 for one exemplary temperature. Thetemperature dependence of the FWHM is displayed in Fig. 18. Like for the crossoverfield, we find the data sets of the different samples to be in excellent agreement. Thisemphasizes that the Hall crossover is robust against sample dependencies whereasthe background contribution and the absolute values of the Hall coefficient are in-deed sample dependent. In addition, the width extracted from the single-field andcrossed-field Hall effect measurements and from the analogous crossover in the lon-gitudinal magnetoresistivity match very well. This indicates that we probe the sameFermi surface change with both the Hall and the magnetoresistivity measurement.

As already deduced from the RH(B2) curves, the width of the crossover shrinksas the temperature is reduced. This width reveals a fundamental property of thequantum critical Hall crossover: Taking all these data sets together, the temper-ature dependent width of the crossover is best described by a proportionality totemperature, see the solid line in Fig. 18. No signature is observed in FWHM(T ) at


46

0.0 0.1 0.2 0.30.0

0.1

0.2

0.3

FW

HM

(T

)

T (K)

YbRh2Si

2 crossed- single- magneto- field field resistance

RH(B

2) R

H(B

1) ρ(B

2)

sample 1 sample 2

~

TN

Figure 18. Vanishing width of the Hall crossover in YbRh2Si2 after Ref. [53]. The full-width at half-maximum (FWHM) was extracted from the derivative of the fits of eq. (47) to RH(B2) in the crossed-field

experiment (closed symbols), to RH(B1) of the single-field Hall data (open symbols) and magnetoresistance(crosses). Inset of Fig. 14 illustrates the definition of FWHM. The values of the single-field experimentwere scaled by 1/11 to account for the magnetic anisotropy. Solid line marks a linear fit to all data sets upto 1 K which intersects the ordinate at the origin within experimental accuracy [49]. Vertical arrow marksthe Neel temperature. Reproduced from S. Friedemann et al., Proceedings of the National Academy ofScience USA 107 (2010) 14547 [53].

TN. This is in contrast to R0H and thus also the height which changes to a quadratic

temperature dependence below TN. A tendency towards saturation of FWHM(T )seems to set in for the crossed-field Hall and longitudinal magnetoresistivity databelow 30 mK whereas the single-field Hall-effect data continue the decrease linearlydown to the lowest temperature accessed. It is most likely that the crossed-fieldHall data are affected by the proximity to the classical phase transition as at theselowest temperatures the Hall crossover significantly interferes with the classicalphase transition at TN(B). This can be seen from the fact that at lowest tempera-tures the FWHM extends substantially into the ordered phase (cf. horizontal barsin Fig. 16). For the single-field Hall experiment the classical fluctuations should besmaller according to this reasoning. Such a difference may well originate from thedifferent orientation of the tuning field for the two experiments. For the crossed-field Hall experiment with the tuning field in the magnetically easy plane, themagnetization and thus the classical magnetic fluctuations are almost one orderof magnitude larger compared to the single-field experiment with the tuning field(B1) along the magnetic hard axis. Nevertheless, within the experimental accuracyall data agree with the fit down to the lowest temperatures. Consequently, theFWHM extrapolates to zero in the limit T → 0. Together with the crossover fieldB0 converging to the QCP and the step height remaining finite, the vanishing widthsuggests that the crossover in the Hall coefficient transforms into a discontinuityat the QCP. This indicates a reconstruction of the Fermi surface.

The single-field and crossed-field Hall experiments reveal a critical crossover inthe Hall coefficient which matches in its position and width. This is in accordancewith the considerations presented in section 4.2. In particular it reflects that ac-cording to eq. (38) the single-field Hall results can be decomposed into an orbitalcontribution and a Zeeman contribution; the former is expected to be identical tothe result of the crossed-field Hall experiment. Only the step height appears to bedifferent for the two experiments which should than be associated with the Zeemanterm. It remains to be understood in detail how the step height is changed. Onepossibility is that it may originate from the different field scales involved in thetwo experiments.

A change of the scattering rates of the two dominating bands crossing the Fermi


47

level [187] can be ruled out as the origin of a jump in the Hall coefficient. Thesample dependencies associated with the background contribution are explainedincorporating different relative scattering rates of the two bands which arise fromminute differences in the composition. The Hall crossover however originates fromthe quantum criticality. Furthermore, the tuning parameter is varied continuouslyand tunes the material through a continuous phase transition. A change in thescattering rates at the latter can thus not lead to a discontinuous evolution of theHall coefficient.

The abrupt change of the Hall coefficient is associated with a Fermi surfacecollapse which is incompatible with the expectations for an SDW QCP. For an SDWQCP a smooth evolution is expected theoretically [28, 47, 48]. The Fermi surfaceof a SDW state is reconstructed from that of the paramagnetic state through aband folding. When the SDW order parameter is adiabatically switched off, thefolded Fermi surface is smoothly connected to the paramagnetic one. As a result,the Hall coefficient does not show a jump provided the nesting is not perfect [48].In fact, for the prototypical material of itinerant antiferromagnetism, elementalChromium, the Hall coefficient seems to evolve smoothly through the QCP. Thisis seen from studies in which the magnetism is suppressed by hydrostatic pressureor through a combination of Vanadium substitution with pressure [45, 46].

We can rule out that the discontinuity in the Hall effect is related to a breakdownof the low-field limit as it might occur at a field induced SDW QCP. It was pointedout theoretically that for a magnetic-field driven QCP the non-zero critical fieldcan give rise to a small discontinuity in the magnetotransport coefficients even inthe case of an SDW QCP [50] (see section 3.4.2). If the energy scale of the vanishingmagnetic order becomes smaller than the energy scale associated with the Lorentzforce the weak field limit breaks down and this may lead to a non-linearity inmagnetotransport. In the Hall coefficient the non-linearities of the conductivitiescan add to a jump which would be smeared by disorder. Up to date, however, sucha jump in RH at an SDW QCP has not been seen in any other quantum criticalsystem. For the case of YbRh2Si2 we exclude this effect by using the crossed-field geometry. This setup allows to apply a vanishingly small Hall field B1 whilethe material is tuned to its QCP via a critical value of the field B2. Thus, theenergy scale associated with the Lorentz force (arising from B1) is negligibly smallavoiding the above described breakdown of low-field magnetotransport. In fact, theHall resistivity remains linear even at the critical value of the tuning field B2 ascan be seen from Fig. 12. In addition, the signatures due to low-field breakdownare expected to change with residual resistivity. Consequently, the fact that theHall crossover is observed for two samples with residual resistivities differing bya factor two discards this scenario. Furthermore, with the width and position ofthe crossover of the single-field experiment being identical to the crossed-field Hallresults even the single-field results seem to be unrelated with a breakdown of low-field magnetotransport.

The interpretation of the results from Hall effect measurements in terms ofa Kondo breakdown are corroborated by thermal transport [10] and thermody-namic measurements [189]. In particular, the isothermal, linear magnetostriction,λ[110](B, T ) = ∂ lnL/∂B (where L is the length along the [110] direction and B⊥c),has been measured down to temperatures of 20 mK. Interestingly, λ[110](B, T ) canbe fit by a crossover function analogous to eq. (47) and yields crossover field valuesBλ

0 in good agreement with those obtained from Hall effect measurements [189].We note that all the low-temperature measurements on YbRh2Si2 so far (including,e.g., magnetostriction, heat capacity and susceptibility) indicated a second orderphase transition at B0.


48

Very recently, Hall effect measurements at high magnetic fields have been con-ducted on YbRh2Si2 [190]. Beyond 12 T, the Hall resistivity ρxy is independent oftemperature within the measured range 0.05 K ≤ T ≤ 2.75 K. This is consistentwith a suppression of the Kondo state at these high fields. The Hall coefficientRH exhibits several small features which are in good agreement with changes inthe topology of the calculated Fermi surface [191]. The overall increase of RH inthe field range between about 6 − 10 T may signal the de-renormalization of theKondo quasiparticles. Most importantly, renormalized band structure calculations[191] indicate the crossing of the Fermi level of one of the spin-split bands at above10 T which is in line with our Hall results, and a Lifshitz transition at these highfields. We emphasize that—in contrast to these results at high magnetic fields—aLifshitz transition at small fields B0, i.e. close to the QCP, is unlikely, specificallyin view of the observed linear increase of the width of the Hall with temperature,FWHM ∝ T , cf. Fig. 18.

Clearly, it is desirable to support the results of the Hall measurements by other,possibly more direct, probes of the Fermi surface in YbRh2Si2, like ARPES ordHvA effect measurements. However, given the low temperatures and small mag-netic fields at which consequences of the QCP are expected to be observed a con-firmation of the Kondo breakdown scenario by these methods is not suitable atpresent. Nonetheless, ARPES measurements have been conducted down to about10 K in zero magnetic field and clearly demonstrated the (dynamical) hybridizationbetween Yb 4f and conduction electrons [192]. Moreover, dHvA effect measure-ments [193] indicated the presence of a large Fermi surface at about 8 T, i.e., inthe Fermi liquid regime of Fig. 1.

5.2. Comparison to other candidates for Kondo breakdown

A continuous evolution of the Fermi surface was identified in CeRu2Si2 with thehelp of Hall effect measurements [194]. In fact, dHvA measurements suggested alocalization of f electrons at the metamagnetic quantum phase transition in resem-blance to the Kondo breakdown scenario. Hall effect measurements down to below10 mK, however, revealed a Lifshitz transition as the underlying mechanism: Oneof the spin-split branches of a Fermi surface sheet shrinks to a point at the meta-magnetic critical field giving rise to signatures in both the resistivity and the Hallresistivity. At high temperatures a pronounced maximum is observed in the Hallresistivity which shrinks as the temperature is lowered and transforms to a kinkat lowest temperatures. Similar behavior might be present in UPt3 where only amaximum at elevated temperatures was observed [82]. The small kink as observedin CeRu2Si2 would not be resolved in the high-field measurements on UPt3. Impor-tantly, the slope of the Hall resistivity, equivalent to the differential Hall coefficient,approaches identical values in CeRu2Si2 on either side of the transition which ba-sically rules out a Fermi surface reconstruction. This is to be contrasted with thedata for YbRh2Si2 which show different values of the differential Hall coefficient oneither sides of the QCP. In addition, such a Lifshitz transition would lead to drasticchanges in the signatures in the Hall effect upon variation of the residual resistivitywhich is in disagreement with the fact that identical signatures are observed forYbRh2Si2 in two samples with residual resistivities differing by a factor of almost2.

A more complicated Lifshitz transition was recently suggested to explain themeasured Hall effect of YbRh2Si2 [195]. Here, it is assumed that there is a hithertounknown ultra-narrow band in the band structure and that one of its Zeeman-split components sinks below (or above) the Fermi energy by the critical field.


49

Figure 19. (a) Selected isotherms of the differential Hall coefficient as a function of magnetic field forYbAgGe after Ref. [196]. Magnetic field is oriented parallel to the crystallographic c axis. Except forT = 50 mK curves are shifted by 0.1 nΩcm/Oe increments for clarity. Phase diagram for fields perpendicular(b) and parallel (c) to the c axis. LRO denotes the long range magnetically ordered phases. Stars representthe position of the crossover in the differential Hall coefficient and crosses depict the onset and end ofthe Hall crossover, respectively. They define the width of the Hall crossover which is emphasized by theshaded region. NFL and FL correspond to the non-Fermi-liquid and Fermi-liquid regime as deduced fromfurther features in the differential Hall coefficient. Reprinted figure with permission from S.L. Bud’ko etal., Physical Review B 72, 172413 (2005) [196]. Copyright c© (2005) by the American Physical Society.

However, this proposal invokes an unphysically small bandwidth of about 5 µeVand is unlikely to be relevant to YbRh2Si2.

Field induced quantum phase transitions were also studied in two other heavyfermion compounds by means of Hall effect measurements, namely URu2Si2 [127]and YbAgGe [196]. In URu2Si2 the Hall resistivity nicely tracks the different phasestraversed when increasing the magnetic field to beyond 35 T. For each of the phasesa proportionality of the Hall resistivity to the magnetic field corresponding to aconstant Hall coefficient reflects the different Fermi surface configurations. Thetransition between these phases occurs abruptly indicating discontinuous phasetransitions. In fact, the Hall effect shows no signature at the putative QCP under-lying the field-induced phase which surrounds it as a dome. This emphasizes theunique role of YbRh2Si2 in which the QCP is not masked by an emergent phaseproviding the chance to study the Hall effect across the QCP.

In YbAgGe, the differential Hall coefficient shows a crossover like structure sim-ilar to that observed in YbRh2Si2. Exemplary curves are reproduced in Fig. 19(a).The position of the crossover is tracked in the phase diagram shown in Figs. 19(b)and (c) for two different field orientations. It is apparent that the crossover posi-tion is linked to the suppression of long range magnetic order, similar to what wasobserved in YbRh2Si2. However, this seems to be different for the two differentorientations. For the field perpendicular to the crystallographic c direction the po-sition of the Hall crossover seems to converge directly to the magnetic instabilitywhereas for fields along the c axis a finite distance appears to persist in the extrap-olation to zero temperature. This is in contrast to the results on YbRh2Si2 whichshow identical behavior for both field orientations (cf. Fig. 16). The main difference


50

to YbRh2Si2 is, however, that for YbAgGe a finite width of the crossover persistsdown to zero temperature. This is most prominent for fields along the c axis ascan be seen from the finite range of the shaded area at lowest temperatures. Likethe position of the crossover also the width is different for different orientations ofthe magnetic field and can not be scaled by the magnetic anisotropy, in contrastto the findings for YbRh2Si2. In summary, these findings represent more complexbehavior and, in particular, the finite width favors an SDW scenario for YbAgGe.

Quite recently, the Kondo breakdown scenario has been suggested to be applica-ble to the cubic heavy fermion compound Ce3Pd20Si6 [197]. Although the materialcrystallizes in a rather complex structure the magnetic Ce ions are arranged on twocubic lattices and in cage-like environments typical for the clathrates [198]. The re-sults of the single-field Hall effect measurements can nicely be analyzed by eq. (47)indicating similarities to the Kondo breakdown in YbRh2Si2, cf. Fig. 16. In partic-ular, the crossover in the Hall resistivity also appears to sharpen to a discontinuityin the extrapolation to zero temperature indicating a Fermi surface reconstructionat the critical field of the antiferromagnetic state. In contrast to YbRh2Si2, how-ever, the Fermi surface reconstruction does not coincide with the transition into aparamagnetic ground state. Rather, a second ordered phase with higher transitiontemperature and extending to high magnetic field encloses both the antiferromag-netic phase and the Fermi surface reconstruction. The finite-temperature crossoverline associated with the change of the Fermi surface crosses the phase boundary ofthis second phase at finite temperature in the B–T phase diagram of Ce3Pd20Si6.

The different behaviors of YbRh2Si2 and Ce3Pd20Si6 have been discussed in theframework of a global phase diagram in which the degree of quantum fluctuations ofthe local-moment magnetism frustration is plotted against the normalized Kondocoupling [199–201]. Both the quantum fluctuations and the Kondo effect weakenmagnetic order whereas solely the Kondo coupling is predicted to induce a Fermi-surface reconstruction.

Due to its cubic structure Ce3Pd20Si6 is expected to exhibit rather three-dimensional, i.e. small, quantum fluctuations. Therefore, it is naturally settled inthe low-frustration regime of the global phase diagram where the Kondo breakdownoccurs within the ordered phase, as sketched in Fig. 20. However, it is important toscrutinize this picture further by investigating the NFL behavior associated withthe Kondo breakdown and clarifying the nature of the high-temperature orderedphase in Ce3Pd20Si6.

YbRh2Si2 has a tetragonal structure and may, therefore, exhibit more two-dimensional fluctuations. This increase in the influence of fluctuation upon low-ering the dimensionality would place YbRh2Si2 at a somewhat higher frustrationlevel than Ce3Pd20Si6. In this regime of the global phase diagram a coincidenceof the Kondo breakdown and the magnetic-to-paramagnetic transition is predictedjust as observed in YbRh2Si2. The separation of the Kondo breakdown and themagnetic transition observed in YbRh2Si2 under chemical [202] and hydrostatic[203] pressure indicate that in YbRh2Si2 the frustration can be fine-tuned therebychanging its level of frustration and consequently its vertical position in the globalphase diagram.

The hexagonal structure of YbAgGe makes this material a promising candidatefor further increase of frustration. This would settle YbAgGe in the regime ofthe global phase diagram where the magnetism is suppressed before the Kondobreakdown is reached. The fact that the signature in the Hall effect for fields parallelto the c-direction of YbAgGe is separated from the magnetically ordered phasecould be in accordance with this prediction. However, a more detailed analysis ofthe temperature evolution of the signatures in the Hall effect may possibly allow


51

Figure 20. Global phase diagram for heavy-fermion systems. Both magnetic frustration (G) and Kondocoupling (JK) suppress the antiferromagnetic (AF) ground state giving way to a paramagnetic phase (P).The Fermi surface reconstruction associated with the breakdown of the Kondo effect marks the crossoverfrom the large Fermi surface (subscript L) which includes the f electrons to the small Fermi surface(subscript S) excluding f electrons. The heavy-fermion compounds YbAgGe, YbRh2Si2, and Ce3Pd20Si6are placed according to the observed relation between magnetic order and signatures in the Hall effect.This is justified by structural arguments (see text) giving rise to different levels of quantum frustration.Figure based on Q. Si, Phys. Stat. Solidi B 247, p. 476, 2010 [201]. Copyright (2010) by the WILEY-VCH.

to scrutinize whether this system undergoes a Fermi surface reconstruction.

5.3. Hall effect and scaling behavior

The proportionality to temperature of the FWHM of the Hall crossover inYbRh2Si2, cf. Fig. 18, allows further insight into the nature of the quantum criti-cality. In order to understand the relevance of this proportionality we have to recallwhat the Hall coefficient is sensitive to. For the given Fermi liquid ground states oneither side of the QCP, the Hall coefficient probes the Fermi surface. R0

H is assignedto the small Fermi surface on the low-field side, and R∞H is assigned to the largeFermi surface on the high-field side in agreement with renormalized band structurecalculations [187]. At zero temperature, the transition from one Fermi surface tothe other is sharp leading to a step in the Hall coefficient. This is in accordance withthe experimental observation of a vanishing FWHM. At finite temperature—in theabsence of a phase transition—a crossover from one Fermi surface to the other isexpected. This is reflected by the crossover in the Hall coefficient. The width ofthis crossover is determined by the single-electron excitations across the Fermi sur-face. In the quantum critical regime they obey a linear-in-temperature relaxationrate as a result of the underlying E/T scaling expected for the Kondo-breakdownQCP [53]. These relaxation processes allow a continuous connection of the differentFermi surfaces at finite temperatures. As the Hall crossover appears to mirror theFermi surface crossover, the linear-in-temperature FWHM hints towards an linear-in-temperature relaxation rate of the single electron states and thus towards anenergy over temperature scaling of the electronic excitations [53]. A conventional(3D SDW) QCP, by contrast, is predicted to obey E/T x scaling with x > 1 whichwould lead to a super-linear T dependence of both the relaxation rate and the Hallcrossover width.

The systematic study of the Hall effect and magnetoresistivity in YbRh2Si2 pro-vides currently the best evidence for a Fermi surface reconstruction at a QCP.The discontinuity of the Hall coefficient in the limit of zero temperature re-flects this change of the electronic structure. Moreover, the scaling analysis of theHall crossover width at finite temperatures sheds further light on the unconven-tional nature of the quantum criticality in YbRh2Si2. The proportionality of theFWHM to temperature observed over almost two decades hints towards a linear-


52

in-temperature relaxation rate of the single electron states which itself is linkedto E/T scaling of the electronic excitations. Consequently, YbRh2Si2 seems to bethe first material where both the fundamental signatures of the breakdown of theKondo effect, a collapse of the Fermi surface and an E/T scaling of the criticallyfluctuating Fermi surface have been observed. This indicates that the macroscopicscale-invariant fluctuations arise from the microscopic many-body excitations as-sociated with the collapsing Fermi surface. The linkage between microscopics andmacroscopics provided by the fundamental E/T scaling is expected to be a key-stone to understand the physics of correlated materials. This might apply to awide class of materials with abnormal finite temperature behavior for which theevolution of the Fermi surface as a function of a control parameter plays a centralrole like, e.g., in the case of the high-temperature cuprate superconductors.

The crossover in the Hall effect defines a new energy scale called T ?(B) whichis confirmed by other transport measurements as well as by thermodynamic mea-surements [189]. This energy scale vanishing at the QCP is reminiscent of theadditional local energy scale proposed in the Kondo breakdown which relates tothe break up of the quasiparticles. Interestingly, sufficient changes in the unit cellvolume lead to a detachment of the antiferromagnetic QCP from the T ?(B) line.This was realized by isoelectronic substitution on either Rh [202] or Si [204] sites,hydrostatic pressure [203] or a combination of hydrostatic and chemical pressure[205]. For lattice expansion, the T ?(B) line is separated from the magnetic QCPwith the intermediate regime featuring non-Fermi-liquid behavior in a finite rangeof the tuning parameter. It still has to be tested whether the scaling behavior ofthe Hall crossover persists under lattice changes which might help to understandthe nature of the emergent non-Fermi-liquid phase. While the existing experimentshave indicated a finite range of chemical pressure in which the two concur [206],this issue deserves further investigation, especially under external pressure. It isalso important to examine how this relates to the unconventional critical scalingseen in both the specific heat and Hall effect [207].

6. Hall effect in systems with 115 type of structure

6.1. Influence of magnetic fluctuations on superconductivity

As stated in section 2.5, a remarkable manifestation of quantum many body physicshas been the gradual realization that the apparently inimical phenomena of super-conductivity and magnetism could be intimately related. At face value, the coex-istence of these two diverse electronic ground states seems unlikely, since the pre-requisites for these states (itinerant and localized electrons, respectively) appearto be contradictory. Moreover, it was also shown [208] that even small amountsof magnetic impurities were detrimental for the stability of the superconductingcondensate in the conventional (BCS) superconductors known at the time. Exper-imentally, this was supported by early investigations using binary rare earth alloysof the form La1−xRx, (R denotes other members of the Lanthanide series), wherethe suppression of superconductivity in La appeared to be directly correlated withthe 4f spin configuration of the dopant [209]. The effect of impurities has beentreated within the theory by Abrikosov and Gor’kov [210]. At a microscopic level,the drastic suppression of superconductivity due to the presence of a magnetic en-tity can be understood to occur as a result of a “pair-breaking” interaction thatarises from the interaction of the magnetic moment (µ) with the spin of the con-duction electron (via the Zeeman interaction) or that of its vector potential withthe momenta (ppp) of the electrons. The influence of an internal magnetic field gen-


53

erated by a system with (or close to) magnetic order on a BCS superconductor wasexamined by Berk and Schrieffer [211], who showed that strong (ferro)magneticexchange would suppress the superconducting ground state. Using the exampleof Pd, which was known to have an exchange-enhanced spin susceptibility, it wassuggested that only a non-superconducting state was feasible for reasonable valuesof phonon interaction strengths [211].

A relatively simple manner in which magnetism and superconductivity can coex-ist is by spatial separation, wherein the magnetic sublattice is relatively decoupledfrom the sea of conduction electrons. This is the case in a variety of systems like theChevrel phases (of the form MxMoX8; where M is a rare earth metal, and X is achalcogenide) [212], the rhodium borides [213, 214], the ruthenocuprates [215] andthe borocarbides [216]. However, the heavy fermion superconductors are distinctin the sense that here magnetism and superconductivity both involve the sameset of f electrons, and it was the discovery of heavy fermion superconductivity inCeCu2Si2 which unambiguously demonstrated that superconductivity was clearlysustainable in an inherently magnetic environment [2]. Moreover, it was also obvi-ous that these strongly renormalized heavy electrons were a necessary condition forsuperconductivity, since the homologous LaCu2Si2 where the 4f electrons, and con-sequently, the heavy fermion state were absent did not exhibit a superconductingground state. Long-range magnetic order is in close vicinity to the superconduc-tivity in CeCu2Si2, as is evidenced by the fact that a magnetically ordered stateemerges in homogeneous CeCu2Si2 samples with very small Cu deficit [217]. Thisrelation between unconventional superconductivity and magnetism is now well rec-ognized, and the heavy fermion systems have turned out to be an interesting test-ing ground where such emergent behavior is observed. Within the same ThCr2Si2structure, a number of other Ce based systems have now exhibited unconventionalsuperconductivity, thought to be mediated by antiferromagnetic fluctuations. Forinstance, CeNi2Ge2 exhibits ambient pressure superconductivity with a Tc ≈ 0.3K [218], whereas ambient-pressure magnetic order gives way to pressure-inducedsuperconductivity in CePd2Si2 (TN ≈ 10 K)and CeRh2Si2 (TN ≈ 36 K) [11, 219].The closely related CeCu2Ge2 was also shown to be superconducting at exter-nal pressures of the order of 7 GPa [220], though the physics of this system, likeCeCu2Si2, is complicated by the presence of two superconducting domes [221] thatpossibly arise from different underlying mechanisms: (i) from a magnetic instabil-ity as in the above mentioned systems, and (ii) from charge density fluctuationsdue to a valence transition in Ce3+ [222, 223]. A number of U-based systems alsoexhibit unconventional superconductivity, notable amongst them being UPt3 [92]and URu2Si2 [93, 94]. In UPt3 a superconducting condensate was seen to emergeat Tc ≈ 0.5 K from within a magnetically ordered ground state which sets in ataround 17.5 K. Though initial analysis of the specific heat was used to specu-late on the possibility of ferromagnetic spin fluctuations in this system [224, 225],the antiferromagnetic nature of these correlations was established by the use ofneutron scattering measurements [226, 227]. A low temperature superconductingground state is also observed to emerge from within an ordered magnetic state inthe hexagonal UPd2Al3 and UNi2Al3, with Tc-values of 2 K and 1.1 K, respectively[95, 228]. A valuable addition to this class of materials has been the CeM In5 (M =Rh, Ir or Co) family of compounds, discussed in detail in the subsequent sections.

In all the systems mentioned above, superconductivity is observed in the pres-ence (or in the vicinity) of long-range antiferromagnetic order. However, in a smallnumber of heavy fermion systems, the superconducting condensate is known toform in the presence (and through the mediation) of ferromagnetic fluctuations.Needless to say, this places more stringent conditions on the symmetry of the su-


54

perconducting order parameter, since the spin-singlet state (in which the pairingquasiparticles have opposite spins) is ruled out. In unconventional superconduc-tors in the presence of strong antiferromagnetic fluctuations, stabilization of thesespin-singlet Cooper pairs with a non-zero angular momentum is most likely in ad-wave state. It is this aspect of heavy fermion superconductivity that binds itwith the superconducting cuprates. However, when Cooper pair formation is me-diated by ferromagnetic fluctuations, the spin-triplet state is most feasible [229] inwhich the Cooper pairs have an effective odd angular momentum (analogous tothat seen in the superfluid 3He phase). The existence of superconductivity in thevicinity of ferromagnetic order was first demonstrated in UGe2, where applicationof external pressures greater than 1 GPa was seen to result in the stabilizationof a low-temperature superconducting ground state from within the ferromagneti-cally ordered state [230]. The orthorhombic URhGe and UCoGe also belong to thesame class, with the superconducting condensate being stabilized from within theferromagnetic ordered state at ambient pressures [231, 232].

6.2. Interplay of magnetism and superconductivity in CeMIn5

The CeM In5 (where M = Co, Rh or Ir) family of heavy fermion systems haveemerged as a focus of intense investigations since the turn of the century. Struc-turally, these systems can be looked upon as layered variants of the CeIn3 system,and can be generally described as CenMmIn3n+2m, where n layers of CeIn3 alter-nate with m layers of M In2 [233]. CeIn3 is an ambient pressure antiferromagnet,with a transition temperature of 10 K. On the application of pressure, non-Fermiliquid behavior emerges near 2 GPa, where long-range antiferromagnetic order issuppressed to zero temperatures, and superconductivity with a maximum Tc ∼ 0.2K emerges [11, 234]. Current interest in the CeM In5 family of systems (equivalentto the n = m = 1 variants of the general form described above) was sparked offwith the observation of pressure-induced superconductivity in CeRhIn5, with a su-perconducting transition temperature Tc ≈ 2.1 K at 2.1 GPa [235]. Subsequently,ambient-pressure superconductivity was discovered in the Ir- and Co-based systems[90, 91]. Superconductivity has also been discovered in the n = 2, m = 1 variants,with Ce2RhIn8 being superconducting at Tc ∼ 1.1 K at 1.6 GPa [236] and thesystems Ce2CoIn8 and Ce2PdIn8 becoming superconducting at p = 0 and belowTc ≈ 0.4 K [237] and Tc = 0.68 K [238], respectively. Recently, the first Ce-based(pressure induced) heavy-fermion superconductor of the type n = 1 and m = 2was discovered: CePt2In7 (Tc = 2.1 at p ≈ 3.1 GPa) [239].

The flurry of attention is primarily due to the intricate interplay between mag-netism and superconductivity which this family of heavy fermion metals havedemonstrated. For instance, CeCoIn5 is an ambient-pressure superconductor with aTc ≈ 2.3 K which is the highest reported value among all Ce based heavy fermionsystems to date. Magnetism lies in close proximity to superconductivity in thissystem, and a magnetic field-induced quantum critical point can be approachedby the application of magnetic fields of the order of the superconducting uppercritical field (µ0Hc2 ≈ 5 T) [240]. Several measurements have indicated that thesuperconducting gap function has line nodes and is most likely to have a d-wavesymmetry [241]. Closely related is the system CeRhIn5, which has an ambientpressure antiferromagnetic transition at about 3.8 K. Though the application ofpressure suppresses long-range antiferromagnetism, access to the quantum criticalpoint is hindered by the onset of superconductivity [235]. The presence of this mag-netic instability has been deduced by specific heat measurements which also pointtowards the existence of a quantum tetracritical point where four distinct phases


55

Figure 21. Phase diagram of the system CeM1−xM ′xIn5 (M,M ′ = Co, Rh, Ir). Reprinted from PhysicaB 312-313, P.G. Pagliuso et al., p.129 [246]. Copyright c© (2002) with permission from Elsevier.

(namely, a non-magnetic phase, a superconducting phase, a magnetically orderedphase, and a region within which superconductivity coexists with magnetic order)meet [42]. The existence of this magnetic instability at 2.3 GPa is also reinforcedby de Haas-van Alphen measurements indicating an abrupt change in the bandstructure [41]. The other ambient-pressure superconductor in this series, namelyCeIrIn5, has remained more enigmatic. For instance, unlike its Co counterpart,superconductivity in this system appears to be well separated from the magneticinstability which in the case of CeIrIn5 is suggested to be metamagnetic in origin[242]. The symmetry of the superconducting gap in CeIrIn5 has also been a mat-ter of dispute. For instance, measurements of thermal conductivity indicated thatthe superconducting gap in CeIrIn5 has a dx2−y2 symmetry [243] which is similarto that seen in CeCoIn5 [244]. This was in contrast to earlier reports which hadascribed the superconducting gap of CeIrIn5 to be a hybrid one with an Eg sym-metry [245]. A dx2−y2 symmetry of the gap in both these systems would imply thatsuperconductivity in both the superconducting members of the Ce-115 family arelikely to be mediated by antiferromagnetic fluctuations in spite of the apparentdifferences in the position of the superconducting regime with respect to the onsetof magnetic order.

The possible difference between superconductivity in the Co and Ir systems isclearly reflected in the phase diagram of solid solutions of CeM1−xM

′xIn5(M,M ′ =

Co, Rh, Ir) as shown in Fig. 21, with a possible superconductor-superconductor crit-ical point at CeRh0.1Ir0.9In5 [246]. This difference is also obvious in doping studieswhich show that antiferromagnetic order emerges cleanly from within the supercon-ducting ground state in CeCo(In1−xCdx)5, whereas these two ground states appearto be separated from each other in the CeIr(In1−xCdx)5 series of systems [247]. Anoverview of some of the physical properties of the members of the CeM In5 familyappears in Ref. [248].


56

CeRhIn5 CeIrIn5 CeCoIn5

Figure 22. Scaling of the relative change in Hall coefficient (ΓHall) as a function of the transformed fieldparameter H∗ for the CeMIn5 family. Reprinted figure with permission from M.F. Hundley et al., PhysicalReview B 70, 035113 (2004) [249]. Copyright c© (2004) by the American Physical Society.

6.3. Hall effect measurements on CeMIn5 systems

6.3.1. Scaling relations in Hall effect

Considering the fact that the CeM In5 systems have been extensively investi-gated in the recent past, it is not surprising that a substantial amount of Hall dataexists in these systems. These systems have also provided overwhelming evidencefor the presence of antiferromagnetic fluctuations, as deduced from Hall effect mea-surements. Work by Hundley and co-workers [249] in the high-temperature regime(2 K ≤ T ≤ 325 K) demonstrated that the field-dependent Hall data in the CeM In5

systems satisfied a scaling relationship that is similar to the one used in analyz-ing the single-impurity magnetoresistance data. Defining the relative change in theHall coefficient as

ΓHall(H) = [RH(H)−RH(H→0)] /RH(H→0) , (48)

it was demonstrated that isotherms of ΓHall plotted as a function of a transformedfield parameter H∗ (defined as H∗ = H/(T+T0)β, where H is the applied magneticfield and T0 and β are scaling parameters) could be made to collapse on top of eachother. These scaled data plots, which were obtained by fixing β = 2 and varying T0

are shown in Fig. 22. This scaling analysis yielded T0 = 4 K, 1 K and 2 K for theRh, Ir and Co systems, respectively. These T0 values are in reasonable agreementwith those estimated for the Kondo temperature from the analysis of specific heatdata.

The measured Hall coefficient of the afore-mentioned systems were also expressedas a sum of a skew scattering term and a second one proportional to the Hallcoefficient in the non-magnetic LaM In5 systems, using a relation:

R(Ce)H (H,T ) = RskewH (T ) + αf (H,T )R

(La)H (T ) . (49)

Here, αf was defined as the f -electron Hall weighting function. It was shown thatαf grows anomalously with the onset of Kondo coherence—the magnitude of whichcould be suppressed by the application of a large magnetic field—demonstratingthe influence of antiferromagnetic fluctuations on the measured Hall response. Thisline of analysis bears some similarity to the two-fluid Kondo lattice model proposedby Nakatsuji and coworkers [250], who modeled the Kondo lattice system into acomponent analogous to the Kondo single impurity phase, and one analogous to acoherent heavy fermion phase, with the temperature evolution of the system beingdetermined by a mixing parameter f(T ). A key aspect is the fact that whereas thecharacteristic energy scale in the Kondo single-impurity regime is the single ionKondo temperature TK , in the heavy fermion fluid it is the intersite coupling inter-action T ∗. Using La dilution on the Ce site of CeCoIn5, it was also demonstrated


57

0.4 0.8 1.26 7 8

0.6

0.7

0.8

78 mK90 mK

0.115 K0.13 K

0.16 K0.18 K0.2 K

0.215 K0.25 K

0.32 K

H / Hmin

0.1 10

10

20

30

μ 0Hm

in (

T)

T (K)

~ T 0.76

90 mK

68 mK0.145 K

b)

μ0H (T)

55 mK

55 mK

0.25 K

0.32 K

| RH

| (

10-9 m

3 / C

)

a)

Figure 23. (a) RH as measured in CeCoIn5. (b) Same data scaled by a normalized field H/Hmin toobtained maximum overlap. The temperature dependence of Hmin is depicted in the inset, with the solidline representing a power law fit. Reprinted figure with permission from S. Singh et al. Physical ReviewLetters 98, p.057001, 2007 [253].

that the condensation of the coherent heavy fermion component is incomplete atthe onset of superconductivity. The need for incorporating these effects in combi-nation was of course realized much earlier, for instance in the analysis of neutronscattering data on the system CeCu6 [251] where a Hamiltonian which accountedfor the influence of both the single-impurity Kondo interaction and the RKKYinteractions was used. The momentum-dependent magnetic scattering observed inthis system was used to infer on the presence of antiferromagnetic correlationswhich could be quenched by the application of a magnetic field. The presence ofthese two contributions and a similar behaviour was also established for the mate-rial CeRu2Si2 [252].

6.3.2. CeCoIn5

A scaling related to one just described in the previous section was also observedin the low-temperature regime (0.05 K ≤ T ≤ 5 K) of CeCoIn5 [253] where it wasshown that the Hall coefficient could be scaled into a single generic curve by nor-malizing their field dependencies by a factor Hmin, as is shown in Fig. 23. Moreover,the Hmin-values which were determined experimentally at low temperatures froma dip in the measured RH (and was thought to represent the inverse of the effectivemobility µeff), could be fit reasonably to a power law, i.e., Hmin = a + bT c. Thispower law fit as shown in the inset of Fig. 23 yielded a = 4.0± 0.7 T, a result thatwas taken to be a signature of the fact that the magnetic instability in this systemand the superconducting upper critical field lie at separate field values. This con-jecture has now been confirmed using measurements of magnetization [254], of thevolume thermal expansion [255], of vortex core dissipation [256], of longitudinalmagnetoresistance [257] and of field-dependent entropy [258] indicating that theputative QCP lies within the superconducting regime. The critical field for thiszero-temperature transition was estimated from the former three types of mea-surement to be about (4.1 ± 0.2) T which is in excellent agreement to the valuededuced from the analysis of the Hall data. This can be seen in the B-T phasediagram Fig. 24(a) which was taken from Ref. [255]. Here, data from our Hall mea-surements, from resistance [259] and magnetoresistance T (ρmax(B)) [240] as wellas from thermal expansion measurements [255] are compiled.


58

Figure 24. (a) B-T phase diagram of CeCoIn5 at ambient pressure. Tcr indicates the change in criticalbehavior [255] and T (ρmax(B)) refers to the position of a maximum in the field-dependent resistance [240].Fermi liquid behavior is inferred below TFL from thermal-expansion measurements in magnetic field [255](filled triangle), Hall effect [253] (open squares) and resistivity measurements [259] (open diamonds). AQCP inside the superconducting (SC) phase (located at the field B+) is inferred. (b) p-B-T phase diagramof CeCoIn5. Shown are results for superconducting transition (black and blue) and Neel temperatures(green) for hydrostatic and negative chemical pressure in CeCoIn5−xCdx to demonstrate the relation ofantiferromagnetism (AF) and superconductivity. Reprinted figure with permission from S. Zaum et al.,Physical Review Letters 106, 087003 (2011) [255]. Copyright c© (2011) by the American Physical Society.

An interesting manifestation of the possible influence of antiferromagnetic fluc-tuations on the Hall response was the observation of a pressure dependent featurein the differential Hall coefficient (RdH = ∂ρxy(H,T )/∂H) in CeCoIn5 [253]. Usingthe same Hmin-values as described in the preceding paragraph, it was shown that|RdH| could be scaled on a curve as shown in Fig. 25. It was suggested that thisdip in |RdH| was related to the influence of antiferromagnetic spin fluctuations ora spin-density-wave-driven gap on the Fermi surface. This suggestion was basedon the fact that this feature was seen to vanish at applied pressures of the orderof 1.2 GPa, which forms the baseline for this curve along with the data at high(H > 1.1Hmin) and low (H < 0.5Hmin) values of the normalized field. This baselinewas thus interpreted to be a reflection of the Fermi liquid regime, and deviationsfrom it were suggested to represent the onset of non Fermi liquid characteristicsarising as a consequence of antiferromagnetic fluctuations. The suppression of thesemagnetic fluctuations by use of pressure in CeCoIn5 has been documented earlier,


59

0.4 0.6 0.8 1.0 1.20.2

0.4

0.6

0.8

215 mK

120 mK

p =

0

H / H

min

| Rd H

| (10

-9 m

3 /C) p

=

1.2

GPa

60 mK

750 mK

Figure 25. Observation of a pressure dependent feature in the differential Hall conductivity in CeCoIn5.The baseline represents Landau Fermi liquid behavior, whereas deviations from it arise due to antiferro-magnetic fluctuations. Reprinted from [253].

and is in reasonable agreement with this scenario [260]. An alternate means ofsuppressing the incipient antiferromagnetic fluctuations in these systems is by theapplication of magnetic fields. This was demonstrated by the combined investiga-tion of the electrical and thermal Hall conductivities (κxy and σxy, respectively) inthe CeCoIn5 system by Onose and coworkers [261]. Separating out the influenceof the electronic contribution (κe) and that due to charge-neutral (spin) excita-tions (κb) to the thermal conductivity it was demonstrated that the latter chargeneutral term displays an anomalous field dependence. It was suggested that thesespin excitations are primarily responsible for the scattering of charge carriers inthe non Fermi liquid regime of the phase diagram. Applied magnetic fields rapidlyquench these excitations, resulting in a rapid enhancement in the mean free path`, which in turn is manifested in the form of novel scaling relationships betweenthe magnetoresistance and the thermal and Hall conductivities.

Besides the above mentioned studies which consider the influence of magneticfluctuations on the scattering of charge carriers, Hall measurements have also beenused to infer a possible modification of the Fermi surface, which then manifestsitself in the form of anomalous transport properties. This aspect will be dealt within section 7, in combination with, and in relation to, Hall effect measurements onthe high temperature superconducting cuprates.

A number of attempts were made to tailor the balance between magnetism andsuperconductivity in the CeCoIn5 system by the use of site-specific dopants. For in-stance, Sn substitution on the Indium site was seen to suppress the superconductingtransition in this system without observable increment in the extent of magneticfluctuations [262]. A novel perspective in this family of systems was gained bythe doping of Cd in the series CeCo(In1−xCdx)5 [247]. Here, magnetism is seento evolve from within the superconducting ground state as a function of increas-ing Cd substitution. Preliminary Hall effect measurements on a member of thisseries CeCo(In0.925Cd0.075)5 revealed an electron dominated Hall response similarto that seen in the parent system [263]. Additional features corresponding to thepossible field-induced destabilization of the antiferromagnetic ground state wasalso observed. Similar observations in the magnetoresistance were then used, inconjunction with neutron diffraction measurements, to chart out the temperature-magnetic field phase diagram in this system [264]. The intertwinned relation ofantiferromagnetism and superconductivity is also suggested by the p-B-T phasediagram, reproduced from [255] in Fig. 24(b). Here, superconducting transition


60

and Neel temperatures are compiled for hydrostatic pressure on CeCoIn5 and neg-ative chemical pressure in CeCoIn5−xCdx.

6.3.3. CeIrIn5

As mentioned in section 6.2, the magnetic instability is known to be well sep-arated from superconductivity in CeIrIn5. Moreover, the magnetic instability ap-pears to be metamagnetic in origin [242, 265, 266]; a factor which modulates thetemperature-magnetic field phase diagram in this material. For instance, unlike thecase of CeCoIn5, the application of an external magnetic field is seen to suppress,rather than stabilize the Fermi liquid ground state. The metamagnetic nature ofthe putative quantum phase transition in CeIrIn5 was also reflected in the scalinganalysis of the Hall effect along the lines mentioned in the preceding section wherethe experimental RH curves within the Kondo coherent regime could be scaledto a generic curve [267]. However, in this case the coefficient yielded a negativevalue, which was ascribed to the fact that there is no magnetic instability in theimmediate vicinity of superconductivity in CeIrIn5. Analysis of the Hall effect wasalso used to deduce the presence of a precursor state to superconductivity in thismaterial [268]. This was accomplished by monitoring the field dependence of theHall angle θH in relation to the inverse of the applied field. Deviations from lin-earity were then used to mark out the existence of a hitherto unidentified statewhich precedes the formation of a superconducting condensate in this system. Theboundary of this precursor state in the temperature-field phase diagram could bescaled onto the boundary of the superconducting regime implying that they couldoriginate from the same underlying phenomena. The existence of this state was inline with prior observations in the related compound CeCoIn5, where a precursorstate to superconductivity was inferred from measurements of the Nernst effect[269] and resistivity under pressure [270].

The anomalous transport properties specifically of CeIrIn5 are discussed in rela-tion to those of the cuprates in section 7.1.2.

6.3.4. Comparative remarks

As discussed in section 2.5 the competition of RKKY and Kondo interactionmay result in quantum critical behavior. Here, a QCP is expected at that valueof the experimental control parameter at which the antiferromagnetic order van-ishes. However, the excessive entropy related to the QCP is often avoided by theformation of superconductivity [cf. Fig. 26(a)]. Nonetheless, the QCP still lurksaround as manifested by non Fermi liquid behavior near the value of the experi-mental parameter at which the QCP is expected but at temperatures high enoughto suppress superconductivity [11, 102].

At temperatures above ∼50 K, the Hall coefficient in CeM In5 with M = Co,Ir, Rh is dominated by skew scattering [249]. Below this temperature, i.e., in arange 2 K . T . 50 K, a T 2 dependence of the cotangent of the Hall angle,cot θH , was interpreted to be caused mainly by the complex band structure of thesecompounds. This was due to the fact that the non-magnetic analogs LaM In5 (withM = Co, Ir and Rh) also exhibited a T 2-dependence of the Hall angle in a similartemperature regime. In systems with complex band structures as has been deducedfor the La-115 systems [271], a temperature-dependent Hall response could arisedue to different temperature dependencies in the contributing electron and holebands, see section 3.4.2. In the magnetic CeM In5 systems, these anomalies wouldbe accentuated due to the addition of strong Kondo and f electron interactions.

At even lower temperatures the results of Hall effect measurements CeCoIn5

could be used to gauge the crossover from non Fermi liquid to Landau Fermiliquid behavior [253], see section 6.3.2 and Figs. 26(b) and (c). It could be shown


61

0.0

0.2

0.41.62.02.4

0 2 4 6 8

M = Rh Co δ

T

a)

liquidnon-Fermi

SCAFM LFL0.1 1

0

10

20

30

40

CoM

=

CeMIn5

T (K)

μ 0H

min (

T)

b)M

=

Ir

1 2T (K)

μ0H (T)

T (

K)

QCP

Hc2 Hmin

LFLc)

CeCoIn5

QCP

-10 0 10 20 300.0

0.4

0.8

1.2 Hc2

T (

K)

μ0H (T)

Hmin

d)

CeIrIn5

Figure 26. (a) Schematic phase diagram of an archetypical heavy fermion superconductor (SC). Theantiferromagnetic (AFM) and Landau Fermi liquid phases as well as the quantum critical point (QCP)are indicated. The positions of CeMIn5 (M = Co, Rh) in their ground states are sketched out. CeIrIn5

cannot be placed within this schematic due to the first-order nature of its magnetic instability. δ denotesthe experimental control parameter which can be magnetic field or pressure. (b) Comparison of the scalingfields Hmin for CeCoIn5 and CeIrIn5, see Figs. 23, 25. Note the break in the temperature scale. (c) H−Tphase diagram of CeCoIn5 with the QCP at around 4 T. (d) Phase diagram of CeIrIn5 without indicationof a QCP.

that the latter behavior is recovered once the antiferromagnetic spin fluctuationsare suppressed either by pressure or magnetic field. As outlined in section 6.3.2,an extrapolation of these results indicated a QCP at around 4 T. Consequently,the quantum critical field is smaller than Hc2. This important result renders theresulting phase diagram Fig. 26(c) to be in good qualitative agreement with thegeneric one, Fig. 26(a), if the zero field state of CeCoIn5 is assumed to be locatedat a position marked by the arrow in the latter.

Applying the same scaling procedure of the Hall coefficient as outlined forCeCoIn5 results in a phase diagram for CeIrIn5 as presented in Fig. 26(d), cf.also section 6.3.3. Clearly, there is no physical meaning of an extrapolation to anegative quantum critical field. But such an extrapolation reveals the close affin-ity between these two members of the CeM In5 family: even though there is noindication for a QCP in CeIrIn5 the zero-field position within the generic phasediagram as marked by the arrow in Fig. 26(a) suggests that the Ir compound nicelyfits into this picture. Such a comparison is corroborated by the presence of strongantiferromagnetic fluctuations [272] in both compounds and similarities in super-conductivity [243, 244] and Fermi surfaces [248, 273]. This implies that the balancebetween Kondo and RKKY interaction (section 2.5) is shifted towards the former


62

in case of the Ir compound. This notion is supported by the ratio TRKKY/TK beingsmaller for Ir (≈ 7) if compared to the Co case (≈ 25). Within these considerationseven CeRhIn5—which exhibits an antiferromagnetic ground state and supercon-ducts upon application of pressure [42]—fits into the picture (TRKKY/TK ≈ 130).It should be noted, however, that Fig. 26(a) does not imply that changing the ele-ment M in CeM In5, application of pressure and magnetic field are interchangeableparameters; rather the arrows only indicate the zero-field ground state propertiesof the respective material.

7. Comparison to Hall effect of other correlated materials

7.1. Copper oxide superconductors and related systems

7.1.1. Cuprates

A discussion of the Hall effect in the heavy fermion metals would not be com-plete without taking note of how measurements of the Hall effect has influencedthe understanding of other strongly correlated electron systems, especially of thehigh temperature superconducting cuprates. A priori, there appears to be very lit-tle in common between these two classes of materials: the heavy fermion metalsare inherently metallic systems with a dense array of magnetic ions, whereas thecuprates are doped Mott insulators. However, the emergence of a superconductingcondensate in both sets of systems has revealed a number of striking parallels. Forinstance, the inherently quasi–2D nature of electronic correlations, the proximityof long range antiferromagnetism (or a spin density wave) to superconductivity,a scaling of the observed transition temperatures with the Fermi energy, and theunconventional (possibly d wave) nature of superconductivity in these materialsall suggest an underlying similarity in the physics of these materials. A welcomeaddition to this narrative has been the recent discovery of superconductivity in theoxy-pnictides and related systems.

The discovery of superconductivity in La2−xBaxCuO4 by Bednorz and Muller[274] fuelled an unprecedented interest into many families of the layered cupratesin an attempt to understand the mechanism of superconductivity, as well as tosynthesize materials with increasing high superconducting transition temperaturesTc. Initial Hall effect measurements in these systems were aimed at discerning howthese systems differed from conventional metals, and to monitor the sign and evo-lution of the charge carriers as a function of temperature and doping. Subsequentavailability of high-quality single crystalline specimens stirred up the utility of theHall effect as a probe of the Fermi Surface topology.

In a simple (uncorrelated) metal, where the electron relaxation times are isotropicon all points of the Fermi surface, the Hall coefficient RH would be expected tobe independent of temperature. This scenario was thought to be valid for reason-ably high temperatures (T ≥ θD, with θD being the Debye temperature) wherephonons were the dominant scattering mechanism which ensured a reasonablyisotropic electron scattering [275]. In practice, RH was seen to be weakly depen-dent on temperature above a characteristic temperature T = sθD, where s variesfrom 0.2 to 0.4 in different metals like Cu, Ag, Cd, and Mg [276–278]. Below thischaracteristic temperature, the observed temperature dependence was thought toarise from an increasing anisotropy in an otherwise isotropic Fermi Surface due toresidual impurity scattering. Not surprisingly, early Hall measurements on the holedoped cuprates indicated that the scenario in this class of materials was dramat-ically different. Measurements on La2−xSrxCuO4 [279], YBa2Cu3O7−δ [280–283]


63

Figure 27. Temperature dependence of the Hall coefficient RH as measured in La2−xSrxCuO4 exem-plifying the temperature dependence of the Hall effect in the hole doped cuprates. Reprinted figure withpermission from H.Y. Hwang et al., Physical Review Letters 72, 2636 (1994) [285]. Copyright c© (1994) bythe American Physical Society.

and Tl2Ca2Ba2Cu3Ox [284] family of systems exhibited a pronounced dependenceof the Hall effect on both the temperature as well as the extent of hole doping.Typically, the Hall voltage was seen to be positive, Fig. 27, which at least in thecase of La2−xSrxCuO4 was thought to arise from the formation of hole pocketsat the corners of the first Brillioun Zone. It was suggested that the area of thesepockets increased with increasing hole concentration, which also accounted for thedoping dependence of RH .

However, experimental signatures like monotonically increasing RH with decreas-ing temperature in all the measured cuprates could not be understood within thesimple picture based on electron-phonon scattering. For instance, in YBa2Cu3O7

the temperature dependence of RH was seen to extend up to 360 K [286]. A sim-ple electron-phonon scattering mechanism as described for simple metals wouldthen warrant an effective θD > 900 K, whereas estimates of θD from heat capacitymeasurements indicated a value which was of the order of 400 K. Measurementson thin films of YBa2Cu3O6+δ indicated that this temperature dependence of theHall effect continued up to about 700 K [287]. A logical extension was the use of atwo-band picture in which the Hall effect from the electrons and the holes partiallycancel each other. Thus, the two band Hall effect could be given as (cf. eq. (4) onp. 13)

RH =µh − µe

en(µh + µe)(50)

where µh and µe refer to the hole and electron mobilities, respectively. In thiscase, the observed net RH could have a temperature dependence if µh and µehave different temperature dependencies, and there existed two nearly symmetricbands which fulfil the above equation in a wide range of temperature [288, 289].However, the fact that this temperature dependence of RH was seen in structurallydiverse systems (with different associated Fermi surfaces) like YBa2Cu3O7 and


64

Tl2Ca2Ba2Cu3O8, would mean that this rather delicate balance had to exist in allthese cuprates in a large part of their phase diagram. The relatively weak pressuredependence of RH as measured in YBa2Cu3O7 [290] would also warrant that µhand µe have similar pressure dependencies as well. Thus, it was evident that thetwo band scenario was implausible.

Another potential mechanism for the observed temperature dependence in RHcould have been the magnetic skew scattering, described earlier in section 2.3.1.If scattering from the magnetic impurities were relatively independent from eachother, the anomalous part of the Hall effect could scale with the (temperaturedependent) paramagnetic magnetic susceptibility χ. The validity of this scenariowas tested by extending Hall measurements to higher fields, where the Zeemanenergy gµBB was clearly higher than the thermal energy kBT . Here, g refers tothe g-factor, µB is the Bohr Magneton, B is the applied magnetic field, kB is theBoltzmann constant. At large values of B, i.e. B > kBT/gµB, the paramagneticsusceptibility (and thus the anomalous Hall effect) should saturate, and should beclearly discernible in field sweep experiments. However, it was observed that inLa2−xSrxCuO4, the Hall voltage as measured at 52 K remained linear up to fieldsof the order of 12 T. Measurements on the electron doped system Nd2−xCexCuO4

indicated that this linear behaviour in the Hall voltage persisted up to 20 T evenat temperatures as low as 1.2 K, whereas a typical g-factor value of 2, at T = 2 Kshould have resulted in a saturation in χ at moderate fields of the order of 1.5 T.

These anomalous transport properties where the resistivity and the Hall effecthad different temperature dependencies could not be accounted for using either theconventional Boltzmann-Bloch models or by incorporating magnetic skew scatter-ing. An additional mystery in the cuprates was the pronounced sensitivity of theHall effect to in-plane disorder. For instance, the addition of Zn or Co impuritiesin the CuO2 planes in YBa2Cu3O7 was seen to result in the suppression of the“Hall slope” defined as d(1/eRH)/dT [291]. An important advance in explainingthis anomalous behaviour in the cuprates was Anderson’s conjecture that thereexist two transport relaxation times in the cuprates which independently influencethe Hall effect and the resistivity in these systems [27]. This was in response tothe observation that the Hall angle θH , as defined in eq. (3), has an unambiguousT 2-dependence in an extended temperature regime, and appeared to be an intrin-sic parameter in this class of materials. In normal metals, there is one transportrelaxation time τtr for electrons on all parts of the Fermi surface, which governsboth the resistivity and the Hall effect. Thus the conductivity σxx is proportionalto τtr and has a T−1-dependence (since ~/τtr ∝ kBT ), whereas the Hall conduc-tivity σxy is proportional to τ2

tr, and thus has a T−2-dependence. However, using aLuttinger liquid formalism, Anderson suggested that transport in these materialswere governed by quasiparticles (spinons and holons) which carry the spin andcharge degrees of freedom, and which independently influence the Hall effect andresistivity. According to this picture, the Hall effect is governed by the transverserelaxation rate τH which is determined by scattering between spin excitations andhas a T−2-dependence, whereas σxy is proportional to τtr · tH , and thus has aT−3-dependence.

More importantly, the Hall angle θH of eq. (3) relies on τH alone and is thus aquantity of fundamental interest. Moreover, it has a T 2 dependence, and is indepen-dent of the impurity scattering which only adds a temperature independent term toboth the scattering rates. This, being an easily verifiable prediction, was confirmedon Zn substituted YBa2Cu3O7 [24], as is shown in Fig. 28. This T 2 dependence ofthe Hall angle was shown to persist up to temperatures as large as 500 K in oxygenreduced samples of the same series by Harris and co-workers [292]. Measurements of


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Figure 28. Temperature dependence of the Hall angle as measured in a series of Zn doped YBa2Cu3O7

samples demonstrating the T 2 dependence of θH independent of the impurity concentration. Reprintedfigure with permission from T.R. Chien, Z.Z. Wang, and N.P. Ong, Physical Review Letters 67, 2088(1991) [24]. Copyright c© (1991) by the American Physical Society.

the optical conductivity confirmed the non-Drude nature of charge dynamics in thinfilms of YBa2Cu3O7 and could be qualitatively explained on the basis of the twodistinct scattering rates [293]. The T 2 dependence of the Hall angle is now known tobe endemic to many cuprate superconductors and was used extensively to empha-size on the anomalous transport properties of many cuprate families. Prominentexamples include the case of Zn, Co, Fe and Pr substituted YBa2Cu3O7 [24, 294–296], Y, Ni and Zn substituted Bi2Sr2CaCu2O8 [297], and Fe, Co, Ni, Zn and Gasubstituted La1.85Sr0.15CuO4 [298]. An interesting manifestation of the existenceof two time scales was the reformulation of the Kohler’s scaling rule in terms ofthe Hall angle. In conventional metals, the Kohler’s rule mandates that the orbitalcontribution to magnetoresistance [ρxx(H)−ρxx(0)]/ρxx(0) can be scaled as a func-tion of the term H/ρxx(0). Harris and co-workers demonstrated that this scalingclearly breaks down in the case of YBa2Cu3O7 and La2−xSrxCuO4 [134]. They alsoshowed that the temperature dependence of the magnetoresistance varies with thesquare of the Hall angle, which is in line with the scenario of two scattering relax-ation rates. A phenomenological transport equation was developed [299, 300] forthe cuprates which allowed distinct scattering rates for τtr and τH as envisaged inthe spin-charge separation scenario. This reproduced the violation of Kohler’s ruleand made predictions for the thermopower [301] but required the two scatteringrates to differ significantly in magnitude which was later shown not to be consistentwith optical transport [293].

Besides the spin charge separation scenario, a prime candidate in explainingthe anomalous transport properties in the cuprates is the nearly antiferromagneticFermi liquid (NAFFL) theory as described in a series of papers by Pines and others[26, 302–306]. This model relies on an anisotropic reconstruction of the (otherwiseisotropic) Fermi surface in the presence of (antiferro-)magnetic spin fluctuations.This is achieved by the formation of “hot” spots and “cold” regions on the Fermisurface, with the hot spots being located at positions in the momentum spacewhere the antiferromagnetic Brillouin Zone intersects the Fermi Surface. All thetransport properties would thus be normalized with respect to these (modified)scattering rates. It was suggested [26] that the Hall effect in these NAFFL systems


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could be broadly classified into three temperature regimes, with σxy varying asT−4, T−3 and T−2 with increasing temperatures. Here, the crossover points be-tween these regimes depend on details of the band structure. In addition to thesetheories, which rely on the Hall angle being linearly dependent on the scatteringrate τH , the marginal Fermi liquid theory by Varma and co-workers relies on asquare scattering response (τ2H) [160, 307–309]. Another avenue of interpretingthe anomalous magnetotransport measurements within the Fermi liquid formalismis the use of current vertex corrections (CVC) championed by Kontani and co-workers [29]. This refers to the current driven by quasiparticles which propagatethough collision events in the Fermi liquid. In a liquid with incipient antiferromag-netic fluctuations, these currents can have a pronounced momentum distributionwhich then correspondingly modifies the observed magnetotransport properties(see, however, section 3.4.2).

Measurements of the Hall effect, in particular the Hall angle, was also used tospeculate on the existence of the pseudogap in some of these cuprates. For instanceAbe and co-workers used deviations from the T 2-behavior in Zn doped YBa2Cu3O7

as a measure of the pseudogap temperature [310]. The direction of change in θHat the onset of the pseudogap was used to suggest that the opening of the gapwas associated with an enhancement of τ−1

H and a reduction in τ−1tr . A similar

deviation of θH at the onset of the pseudogap was also reported in the underdopedYBa2Cu3O6.63 system [17]. The temperature scale of the pseudogap was also usedto scale the Hall number nH and the Hall angle in films of YBa2Cu3O7 indicatingthat the pseudogap is an intrinsic energy scale of these systems [311]. Interestingly,a similar line of analysis was used earlier in crystals of La2−xSrxCuO4 where thetemperature dependent RH was scaled using an arbitrarily defined temperaturescale T ∗, where T ∗ was thought to approximately represent a crossover between atemperature independent to a temperature dependent RH [285, 312].

Fermiology in the Cuprates

The availability of extremely high-quality single-crystal specimens, coupled withsensitive new Hall effect measurements have thrown new light on the nature ofthe Fermi surface in cuprates. Whereas systems in the overdoped regime of thecuprate phase diagram behave like metals to a reasonable approximation, the un-derdoped cuprates are characterized by Fermi surfaces made up of discontinuous“Fermi Arcs” as deduced from spectroscopic measurements [313, 314]. An impor-tant advance was the observation of quantum oscillations as measured in the Hallresistivity in the underdoped YBa2Cu3O6.5, establishing the existence of a contin-uous Fermi surface in this class of systems [315]. The Hall response was observed tobe negative (unlike the typical positive RH observed in other hole-doped cuprates),and this was attributed at that time to the influence of the vortex liquid phasewhich persists in this region of the phase diagram. Moreover, the low frequencyof the observed Shubnikov–de Haas oscillations indicated the presence of a Fermisurface made up of small pockets. Observation of similar oscillations in the mag-netoresistance of YBa2Cu4O8 indicated that the existence of these small pocketswere independent of the details of the band structure and could probably be ageneric feature of many underdoped cuprate superconductors [316]. Monitoringthe sign of the Hall response in the magnetic field regime where these oscillationswere observed, LeBoeuf and co-workers concluded that these small pockets wereelectron like, rather than being hole like [20]. This is surprising considering the holedoped nature of these materials and also because of the fact that the existence of acylindrical, hole-like Fermi surface in the overdoped cuprates has been documented[317]. This could imply that there exists a Fermi surface reconstruction as a func-tion of hole doping, with a possible zero-temperature instability demarcating these


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Figure 29. The violation of Kohler’s scaling (upper panel) and its reformulation in terms of the Hall angle(lower panel) as deduced in CeIrIn5. Reprinted figure with permission from Y. Nakajima et al., PhysicalReview B 77, 214504 (2008) [320]. Copyright c© (2008) by the American Physical Society.

two metallic states. Recently, LeBoeuf and coworkers have reported on a Hall effectinvestigation on the YBa2Cu3Oy system with varying hole doping per planar Cuatom (p) from 0.078 to 0.152 [318]. In the T→0 limit, the Hall coefficient RH wasseen to change from positive to negative, as the doping is reduced below 0.08. Thiswas suggested to arise due to the presence of a Lifshitz transition which modifiesthe Fermi surface topology. The electron pocket which is observed to persist tillthe highest investigated doping level (p = 0.152), was conjectured to arise from anordered state (like some form of stripe order [319]) which breaks the translationalsymmetry of the system.

7.1.2. Comparison of cuprates and heavy-fermion systems

The insight gained in investigating the anomalous transport properties of the un-derdoped cuprates have found resonance in attempting to understand the nature ofpronounced non Fermi liquid characteristics observed in some heavy fermion met-als. This is especially so in the case of the magnetotransport in Ce-115 systems,section 6.3, where an acute interplay between superconductivity and quantum mag-netism has revealed some striking parallels with that observed in the cuprates. Forinstance, in both the CeCoIn5 and CeIrIn5 systems, a quasi-linear temperaturedependence of the resistivity is observed, with this non-Fermi liquid characteris-tic being driven due to the influence of incipient antiferromagnetic fluctuations[90, 91]. The Hall effect as measured in these systems also shows a pronounced


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Figure 30. Modified Kohler’s plots equating the magnetoresistance with the Hall angle as measured inCeRhIn5 at pressures above 2 GPa. Reprinted with permission from Y. Nakajima et al., Journal of thePhysical Society of Japan 76 (2007) 024703 [12]. Copyright (2007) by the Physical Society of Japan.

temperature dependence. More importantly, the Hall angle θH clearly varies as T 2

indicating the disparate nature of scattering processes that drive the resistivity andHall effect in these systems [12, 249, 268, 320–322]. This is also reinforced by theanalysis of the magnetoresistance where the Kohler’s scaling is violated, and thescaling procedure is only seen to work when equated in terms of the Hall angle asis shown in Fig. 29.

In CeIrIn5, an additional anomaly in the form of a breakdown of the modifiedKohler’s scaling is observed very close to the onset of superconductivity. However,this is due to the influence of a precursor state to superconductivity in this system,which influences the resistive and Hall scattering rates in disparate fashions [268].The existence of this precursor state was inferred from the field dependence of theHall angle, and was thought to represent an anisotropic reconstruction of the Fermisurface prior to the formation of the superconducting condensate. In line with priorinvestigations in the cuprates where the energy scale associated with the pseudo-gap was used to scale the resistivity and the Hall effect [311], a model-independentsingle parameter scaling of the Hall angle was observed in CeIrIn5 implying thatthe precursor state represented an intrinsic energy scale of the system [18]. In-terestingly, neither the resistivity nor the Hall effect could be individually scaledusing the precursor state, implying that the onset of this precursor state selec-tively influences the Hall channel alone. This is in consonance with that observedin the underdoped YBa2Cu3O6.63 where the opening of the pseudogap was seen tosignificantly reduce the Hall response leaving the resistivity relatively unaffected[17]. Unlike the systems CeCoIn5 and CeIrIn5, which lie on the paramagnetic sideof the putative Quantum critical point, CeRhIn5 orders antiferromagnetically atambient pressures. Application of external pressure shifts this system towards asuperconducting ground state, and in this regime the pronounced non-Fermi liquidproperties enumerated above are retrieved. Fig. 30 shows the modified Kohler’splot as deduced in CeRhIn5 for a range of pressures above 2 GPa where supercon-ductivity is observed.

Though this similarity between the heavy fermion metals and the superconduct-ing cuprates is surprising, this common ground is possibly brought about by theingredient which influences the electronic properties of both these systems, viz., thepresence of antiferromagnetic fluctuations. In magnetotransport, this is reflectedin the form described above, with the Hall and resistivity being dictated by twodifferent scattering rates. Unfortunately, an analysis of this nature mandates thesimultaneous measurements of the Hall effect and the resistivity which at least inthe case of heavy fermion systems has been scarce. However, it is interesting to notethat for all the systems where such measurements have been reported—namely, the


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Figure 31. Violation of conventional Kohler’s scaling in BaFe2(As0.67P0.33)2. Inset shows the modifiedKohler’s plot with the magnetoresistance equated in terms with the Hall angle. Reprinted figure withpermission from S. Kasahara et al., Physical Review B 81, 184519 (2010) [329]. Copyright c© (2010) bythe American Physical Society.

members of the Ce-115 family and YbRh2Si2 [176]—a T 2-dependence of the Hallangle has been found along with a quasi-linear temperature dependence of resis-tivity. The fact that this behaviour is observed in spite of the wide span of lowtemperature electronic ground states observed in these materials implies that thescenario of two scattering times could be a feature of many heavy fermion systems.

7.1.3. Hall effect in the oxy-pnictides and related systems

A valuable addition to the existing assemblage of strongly correlated electronsystems was the discovery of high temperature superconductivity in an iron basedsystem by Kamihara and co workers [323]. The consequent flurry of activity (fora review see, e.g., [324]) has revealed that emergence of superconductivity in theproximity to a magnetic instability could be a factor which binds these systemswith the heavy fermion systems and the high temperature cuprate superconduc-tors in spite of the different band structures of these systems. Though analysisof Hall effect measurements on these systems have not been extensive till date,there is enough indication at the time of writing this review to indicate the anoma-lous nature of the magnetotransport in this class of materials. For instance, Chengand co-workers reported on the Hall effect and magnetoresistance in single crys-tals of NdFeAsO1−xFx and observed a pronounced temperature dependence of theHall effect [325]. Moreover, analysis of magnetoresistance revealed a breakdown ofKohler’s scaling in this system. The temperature evolution and, more importantly,the sign change of RH in Ba(Fe1−xCox)2As2 was used to conjecture on the natureof the electron-hole asymmetry in this system [326]. Using a two-band Fermi liquidmodel it was inferred that the electrons and holes in this multiband system havedisparate scattering rates. Similar inferences were also drawn by Rullier-Albenqueand coworkers [327]. The use of the two-band Fermi liquid formalism in both theseworks meant, however, that the utility of non-Fermi liquid models were not explic-itly considered.

An interesting pointer in this regard is a recent use of the Hall effect in monitor-ing the doping-induced evolution from a Fermi liquid to non-Fermi liquid regimein BaFe2(As1−xPx)2 [328, 329]. In this series of samples, increasing P substitutionsuppressed the spin-density-wave transition near the end point of which a super-


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conducting regime was uncovered. For the optimally doped system (with x = 0.33),pronounced non-Fermi liquid characteristics like a linear T -dependence of the re-sistivity and a violation of Kohler’s scaling were observed. Moreover, a modifiedKohler’s scaling which again relates the magnetoresistance with the Hall anglewas observed, as shown in Fig. 31. This observation indicates that the anomaloustransport properties in at least some of these multiband systems could mirror thoseobserved in the cuprates and the heavy fermion systems.

7.2. Other systems of related interest

The possibility of the Hall effect and the magnetoresistance having different scat-tering rates has also been discussed in a few other strongly correlated systems. Forinstance, in the prototypical Mott Hubbard system V2−yO3 a pronounced tempera-ture dependence of RH was observed [330]. Moreover, the temperature dependenceof the Hall effect and the magnetoresistance in the paramagnetic regime was dif-ferent; with the Hall angle θH and the resistivity ρxx varying as T 2 and T 1.5,respectively. This was ascribed to the possibility of a zero temperature metal insu-lator transition, which is driven by hole doping in the 3d band as a consequence ofincreasing vanadium deficiency. Another celebrated example is metallic chromiumin which the long-range magnetic order can be driven to absolute zero by alloy-ing with small amounts of vanadium, with the putative QCP being approached atcritical concentrations of about x = 0.035 in the Cr1−xVx system. Magnetotrans-port in specimens of this concentration revealed a temperature dependent RH inconjunction with a T 3-dependence of ρxx and a T 2-dependence of the Hall angleθH [44].

The organics have emerged as in interesting playground in which strong elec-tronic correlations drive a host of novel electronic ground states in close proximityto each other in phase space. Though analysis of the Hall effect in conjunctionwith the magnetoresistance has been relatively scarce in this class of materials, afew examples do exist in current literature. For instance, measurements of mag-netotransport in the superconducting salt κ-(BEDT-TTF)2Cu[N(CN)2]Cl in thepressure range between 4 to 10 kbar reveal a T 2 dependence of the Hall angle θHwhereas the resistivity appears to be sub-linear [331]. Similar behavior was alsoobserved in the system Θ-(DIETS)2[Au(CN)4] where measurements of the Hall ef-fect and magnetoresistance under pressures between 5 to 20 kbar along the c-axisrevealed a linear T -dependence of the resistivity, along with a T 2-dependence of θHover a reasonable temperature span [332]. Considering the fact that these anoma-lous transport properties appear to transcend material classes and is observed insystems as diverse as the cuprates, the heavy fermions, the organics, and othercorrelated metals, we suggest that this disparity between the Hall effect and themagnetoresistance could be a generic feature of many correlated systems in thepresence of strong (antiferro-)magnetic fluctuations.

7.3. Colossal magnetoresistive manganites

A recently highly active field of research in condensed matter physics is concernedwith the manganites, RMnO3 with R being a rare earth, and their doped variants,R1−xAxMnO3 where A denotes a divalent or tetravalent cation. Also in this classof materials the correlations between the electrons play a crucial role [333]. Thesecorrelations involve structural, charge, orbital as well as spin degrees of freedomleading to a vast variety of different types of order that may result in competition,or even coexistence, of different phases. The balance between these phases can be


71

subtle such that small changes of a tuning parameter (again, e.g., chemical compo-sition or magnetic field) prompt a significant response of the material’s properties.Typical examples here are the metal-insulator transition (MIT, at a temperatureTC) and the so-called colossal magnetoresistance (CMR, an unusually strong changeof resistance with applied magnetic field) [334]. Even though the CMR effect withits potential technological applications [335] has originally drawn the research in-terest to the manganites, their rich phase diagrams due to the above-mentionedfour degrees of freedom which gives rise to the possibility for intrinsic inhomogene-ity appears to be of more contemporary interest [336]. The phase coexistence withinhomogeneous electronic or/and magnetic material properties makes the mangan-ites also prime candidates for local probes like scanning tunneling microscopy andspectroscopy (STM/S) or magnetic force microscopy (MFM).

Although the manganites seem to be quite different from the heavy-fermion met-als there are a number of similarities beyond the mere statement of both beingstrongly correlated electron systems [337]. The above-mentioned Kondo effect wasoriginally developed for single magnetic impurities in metals [85]. In the heavyfermion metals, however, the magnetic ions are periodically arranged on the latticeand may actually interact. Therefore, a variant of the Kondo model, often referredto as the “Kondo lattice model” [338, 339], needs to be considered in which the(often antiferro-) magnetic interaction between localized moments and itinerantelectrons at each relevant lattice site is taken into account. On the other hand,the famous double exchange [340] mediates the ferromagnetic coupling betweenMn3+ and Mn4+ ions through two simultaneous hopping events of the eg conduc-tion electrons via the intermediate O2− ion in the manganites. Consequently, thedouble exchange model can be considered as similar to the Kondo lattice modeljust with ferromagnetic coupling of local moments and conduction electrons [341].The combination of spin—resulting from localized electrons—and charge—carriedby conducting electrons—degrees of freedom is also at the base of spintronics [342],and the dilute magnetic semiconductors [343] bear some resemblance to the Kondoimpurity problem. Moreover, the phenomenon of phase separation (in the sense oflocally inhomogeneous electronic or/and magnetic properties) is certainly not onlydiscussed for magnetic oxides (see e.g. [344]) but was applied early on to dopedEu chalcogenides [345, 346], to the thorium phosphide-type structures [347] andto EuB6 [348]. Also, in its “stripe” version it has been considered for cuprate su-perconductors [349, 350] and may even be at play within the coexistence range ofantiferromagnetism and superconductivity in CeCu2Si2 [351].

Hall effect measurements in the manganites were used to infer the Fermi surface ofthese materials [352, 353]. LaMnO3 is an insulating A-type antiferromagnet whosemagnetic properties are brought about by superexchange. The gap in the densityof states (DOS) right at EF is nicely reproduced by band structure calculationswithin the local density approximation (LDA), e.g. in Ref. [354]. Considering asimplified picture of a rigid band structure it is obvious that either removing oradding electrons, i.e. lowering or raising EF with respect to the bands, may result ina conducting material at low temperature. The usual hole doping (e.g. by Ca or Sr)results in a corresponding mixture of Mn4+ and Mn3+ whereas a less well-knownelectron doping (e.g. by Ce) gives rise to Mn2+ and Mn3+. Both types of mixedvalencies allow (within a certain doping range) for Zener double exchange with aferromagnetic metallic ground state of the material. The metallic state prevails upto the ferromagnetic transition at TC where the mobility of the electrons decreasesdue to spin disorder, and the carriers are localized via the formation of Jahn-Teller polarons. Hence, the polaronic insulating state results from the local latticedeformation around Mn3+, and these materials can be considered as polaronic


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δ’

1.0

0.5

0

O-c

onte

nt

4.0

3.0

2.0

Mn4+

Mn3+

Mn2+

1 Ce-doping x = 0 Ca-doping 1

electron doping hole doping

Figure 32. Illustration of the Mn valence resulting from both, the doping on the La site (abscissa) aswell as the oxygen content (ordinate) in La1−xCaxMnO3+δ′ or La1−xCexMnO3+δ′ . The Mn valence is

color-coded in red, blue and green presenting Mn2+, Mn3+ and Mn4+, respectively. Sufficiently high O-excess can easily outweigh an electron doping intended by Ce substitution of La, and may even result inan unwanted Mn3+/Mn4+ mixture.

semiconductors above TC [355, 356].Hall measurements on the manganites are notoriously difficult because of often

large contributions from the (negative) anomalous Hall effect and the magnetore-sistance [333]. Nonetheless, some results of Hall measurements in the more commonhole doped manganites were reported, see e.g. [357] for an excellent overview. Forexample, charge carrier densities of n ≈ 1029 m−3 for La0.65(PbCa)0.35MnO3 atT = 5 K were obtained [358] as expected for a typical metal. At higher temper-atures, sign reversals of RH were observed [359, 360] which could be a result ofdifferent bands contributing to the Hall response, see below. In films there couldbe additional localization effects of charge carriers due to strain or disorder [361].

Another issue becomes specifically obvious if nominally electron doped mangan-ites are considered: the resulting valency of the Mn ions does not only dependupon the A-site doping level x but also on the oxygen stoichiometry δ′ whereLa1−xAxMnO3+δ′ [362]. Electron doping in films La0.7Ce0.3MnO3 grown by pulsedlaser deposition (PLD) at low oxygen partial pressure (. 10 Pa) was evidencednot only by x-ray absorption spectroscopy [363, 364] but was discernible also inHall effect measurements [365]. However, with increasing oxygen content the ma-terial is driven more and more into hole doping [366]. This is sketched in Fig. 32:While oxygen excess (going upward in Fig. 32) is of no harmful consequence tothe nominally hole (e.g. Ca) doped manganites it can easily outweigh or even over-come the intended electron doping through Ce substitution in La1−xCexMnO3+δ′

resulting in an effective Mn3+/Mn4+ mixture [367, 368]. In addition, unwantedphases (e.g. CeO2) are very likely to be found in the electron-doped manganites[366, 368]. The formation of additional phases is certainly also supported by thesmall tolerance factor (t ≈ 0.91 for La0.7Ce0.3MnO3) resulting from the small sizeof the tetravalent ions [369, 370]. The small t may also explain the fact that mostlythin films of La0.7Ce0.3MnO3 or other [371, 372] electron doped manganites couldbe synthesized; with the exception of La0.9Te0.1MnO3 for which electron doping inbulk samples was shown by Hall effect measurements [373]. All these difficultiesmight easily explain the contradicting results found in the literature as much asthey render the electron doped manganites less attractive for further applications.


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As already discussed in section 2.2 an anomalous contribution to the Hall effectis expected in magnetic materials, cf. eq. (2). For several manganites it was foundthat the AHE is negligible at low temperature T TC but contributes significantlyto %xy as T is raised towards TC [357, 365, 374–377]. For La2/3Sr1/3MnO3 andwithin the metallic phase a linear relationship between RS(T ) and %xx(T ) as wellas the temperature driven change of magnetization, M(0) −M(T ), was reported[378]. A similar relation, RS ∝ %αxx, holds in La1−xCaxMnO3 with α = 1.38 forx = 0.3 and α = 0.6 for x = 0.67 where electron-type carriers were observed [353].Below TC the metallic manganites behave as bad metals with the transport beingdominated by holes of low mobility [333, 357, 377]. For comparison we wish to addthat well above TC the bandwidth of the eg electrons is narrowed by the Jahn-Teller distortion along with strong electron-phonon interactions which causes theformation of small polarons [333, 379] whereas manganites in the insulating regimeare governed by variable-range hopping.

Certainly the most interesting temperature regime is the one around the MIT.An early Hall effect study [374] on La1−xCaxMnO3 thin films focusing on thistemperature range again reported RS ∝ %xx as well as the Hall angle (see eq. (3))tan θH ∼M for T > TC. It was pointed out that the large value of RS in the hoppingregime of the manganites is different from the anomalous Hall effect in typicalferromagnetic metals (cf. sections 2.2 and 2.3). While hopping the electrons acquirea phase that reflects their induced alignment to the core spins and is consequentlygoverned by the magnetization [374]. Based on such ideas a model for the AHEwas developed [357, 375] that takes into account the quantal phase due to strongHund’s-rule coupling and the spin-orbit interactions.

We note that an additional (positive) contribution to the AHE is observed inpolycrystalline thin films due to grain boundaries [380].

In layered manganites La2−xSr1+xMn2O7 an additional contribution to the AHEis observed possibly related to the intrinsic inhomogeneity resulting from the spinglass state [381, 382]. In contrast to the manganites, the pyrochlore Tl2Mn2O7

exhibits a negligible AHE and its CMR near TC is governed by a change in chargecarrier density [383]. Similarly, the CMR observed in Eu0.6Ca0.4B6 appears to berelated to a crossover from hole- to electron-dominated transport [384].

8. Summary

The quest of shedding light on such highly complex materials as the strongly corre-lated electron systems calls for a comprehensive study and sophisticated experimen-tal approaches. Amongst the plethora of laboratory based tools routinely employedin the investigation of strongly correlated electron systems, Hall effect measure-ments have proven to be an exemplary one. With other measurement techniquesoften not feasible in the temperature and field range of interest for heavy fermioncompounds, Hall effect measurements can provide insight into the evolution oftheir renomalized Fermi surfaces, even across a quantum critical point. As such,it remains unparalleled in tackling some of the outstanding problems at the fron-tiers of current condensed matter physics. Though demonstrated in the case of aprototypical heavy fermion system, YbRh2Si2 in section 5.1, the inferences drawntherein are crucial for understanding the physics of systems like heavy fermionsuperconductors (in section 6 we focused on those with 115-type of structure) oreven the cuprates, in which magnetism and superconductivity appear to be in-exorably intertwined. The possibility that a similar Fermi surface reconstructionoccurs at the crossover from the overdoped to the underdoped regime in the hightemperature superconducting cuprates continues to be at the focus of extensive


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investigations. A related development pertains to the possibility of two scatteringtimes in the heavy fermion systems as deduced from recent Hall measurementsin some Ce-based heavy fermions—very much in similarity to that observed ear-lier in the cuprates. The recently discovered Fe-based superconductors also appearto share some of these intriguing properties. Considering current issues related tosample purity in these systems, it is not unrealistic to predict that they wouldcontinue to be extensively investigated for several years to come.

In this review we focused on the investigation of Fermi surface effects in heavyfermion metals. Consequently, we concentrated on the normal contribution to theHall effect, leaving the anomalous Hall effect aside, even though it also may yieldvaluable information [67]. Fortunately in this context, the anomalous contributionis often negligibly small for heavy fermion compounds at low temperatures, seesection 2.6. Our material basis also implies that the active research areas of thequantized Hall effect [385], its fractional counterpart [386] as well as the related,topical spin Hall effect [387] and topological phases [388] are not touched upon.

Hall effect measurements probe the electronic states directly at the Fermi energyEF. As a major advantage, such measurements can be conducted over a wide pa-rameter space, e.g. in temperature, magnetic field (including pulsed fields) or pres-sure, making this an invaluable probe. Moreover, the results of Hall measurementscan be compared to theoretical predictions, or could serve as input for theoreti-cal considerations. Here, Hall effect measurements on YbRh2Si2 constitute a goodexample, as outlined in section 5.1. The theoretical prediction and interpretationhave taken advantage of the simplification the zero-temperature limit brings aboutto the evolution of the Hall coefficient across a quantum critical point. At the sametime, we have emphasized that for quantitative understandings as well as for thedescription of the Hall coefficient at finite temperatures, it is often important toincorporate the complexities in the band structure, such as the multiplicity of thebands, or the various interaction effects on the electronic excitations near the Fermisurfaces. Experimentally, it can be challenging to get the samples into the thin,plate-like shape shown in Fig. 3. As many of these materials are good metals thethickness d often needs to be as small as a few tens of micrometers to give reason-ably measurable Hall voltages of several ten nV. In providing detailed informationabout the measurement technique (section 4) as well as some basic theoreticalbackground (section 3) we wish to inspire more research within the exciting fieldof heavy fermion physics and, hopefully, beyond.

9. Appendix

The following list provides references in which results of Hall effect measurementsare presented for numerous heavy fermion and related materials. It is meant toserve as a guide and is by no means exhaustive.

Compound ReferenceCeCu2Si2 [113, 389–392](Ce1−xLax)Cu2Si2 [391]CeRu2Si2 [115, 194]CeRh2Si2 [39]CeNi2Ge2 [393]CeNiGe3 [394]CeAl2 [392, 395, 396]Ce(Al2−xCox) [397]CeSn3 [392]


75

Compound ReferenceCePtSi [119]CeCu6 [114, 121, 389, 398, 399](Ce1−xLax)Cu6 [399]CeB6 [400](Ce1−xLax)B6 [400, 401]CePd3 [113, 392](Ce1−xYx)Pd3 [120]Ce(Pd0.88Ag0.12)3 [113, 392]CeBe13 [113, 392]CeRh3 [392]CeIn3 [402]CeAl3 [115, 403](Ce1−xLax)Al3 [403]CeCoIn5 [12, 249, 253, 261, 321, 322]CeCo(In1−xCdx)5 [263]CeRhIn5 [12, 249, 321, 322]CeIrIn5 [249, 267, 268, 320]Ce3Pd20Si6 [197]Ce2RhIn8 [404]Ce2IrIn8 [404]Ce2CoIn8 [405]Ce2PdIn8 [406]CePt2In7 [407]YbRh2Si2 [49, 53, 176, 186, 190]YbCuAl [113, 408]YbCu2Al2 [113, 408]YbNi2B2C [409]YbAgCu4 [410]YbInAu2 [113, 408]YbAgGe [196]YbPd [408]YbAl2 [408]YbAl3 [408, 411]U2Zn17 [122]URu2Si2 [124, 127, 128]U(Ru0.96Rh0.04)2Si2 [412]UPt3 [82, 115]U2PtC2 [413]UAl2 [115]UGe2 [414]UBe13 [114, 413, 415]UPd2Al3 [416]NpPd5Al2 [417, 418]

Acknowledgments

We thank E. Abrahams, P. Coleman, P. Gegenwart, C. Geibel, S. Kirchner andS. Paschen for insightful discussions. Work at Dresden was partly supported bythe German Research Foundation through DFG Forschergruppe 960. S.N. andS.F. acknowledge support by the Alexander von Humboldt Foundation, Germany.


76 REFERENCES

Q.S. acknowledges the support of NSF Grant No. DMR-1006985 and the RobertA. Welch Foundation Grant No. C-1411.

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